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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS TRANSLATED FROM THE TEXT OF HEIBERG WITH INTRODUCTION AND COMMENTARY BY T. L. HEATH, C.B., Sc.D., SOMBTIMK FELLOW OF TRINmr COLLKGB, CAMBMDGB VOLUME I INTRODUCTION AND BOOKS 1, H * • . • Cambridge : at the University Press 1908 1*1 I THE THIRTEEN BOOKS •OF EUCLID'S ELEMENTS CAMBRIDGE UNIVERSITY PRESS WAREHOUSE C. F. CLAY, Manager. Eiltei: FETTER LANE, E.C. •Mrtml: too. PRINCES STREET. Ikrlte: A. ASHSR AND 00. ; P. A. BROCKHAUS, G. P. PUTNAM'S SON& n» Calcvlte: IfACMILLAN AND CO., Lm [Al/ X^Jk/s merved.] A***r XT rT»« •K(^i«4P4« 'tjV*^ -HmP V^ fttC J? . ^ l^if. iVTi fi- uj Chap. I. . »» II. »> III. >» IV. »» V. »« VI. M VII. t *> VIII. 5 f " IX. CONTENTS VOLUME I. INTRODUCTION. p. eucud and the traditions about him Euclid's other works Greek commentators other than Proclus Proclus and his sources The Text The Scholia Euclid in Arabia PRiNaPAL translations and editions . § I. On the nature op Elements i § 2. Elements anterior to Euclid's . . . : I §3. First principles: Depinitiqns, Postulates AND Axioms . . . ' . i [ § 4. Theorems and Problems . . . . j ( § 5. The formal divisions of a proposition i h § 6. Other technical terms . . . . : § 7. The definitions i [ THE ELEMENTS. I Book I. Definitions, Postulates, Common Notions . ] Notes on Definitions etc. ] Propositions . j Book II. Definitions Note on geometrical algebra Propositions . ^ . . . — •— k X CONTENTS VOLUME II. Book IIL Definitions . Propositions , Book IV, Definitions Propositions , Book V. Introductory hote Definitjons , Propositions , Book VL Introductory hote Definitions . Propositions . Book VIL Definitions Propositions , Book VIIL • 1 Book ;IX, * Greek Index to Vol, H. . FjfGLiSH Ihdex. to Vol. 1L FAOt t ?8 119 Its 138 18S 191 «77 99$ 345 384 431 VOLUME IlL Book X. Introductory note i Definitions 10 Propositions i^ — ^47 . , . , • ,14-101 Definitions il ,/,.... ioi Propositions 4S — 84 iq?-i77 Definitions hi ryj Propositions 85—115 17&-254 Ancient extensions of theory of Book X . 255 Book XL Definitions . * a6o Propositions . - - ' 27a Book XIL Historical note . . < . ' • . . 365 Propositions 369 Book XIII* Historical note 43S Propositions ......*. 44a iPPENDix. L The so-called "Book XI V," (by Hypsicles) 511 IL Note on the so-called **Book XV/* , , 519 iddknda et corrigenda .*...... £21 * Index: Greek 329 English 535 ] n ifir Christoph Schlussel rtad Christoph KUn ERRATA Vol. I. p. i9» line \*i^fcr ** bat not a platfonn and sixpence " rtid " bat not a figare and tizpence" » P> io5» luM lO »• p* 415* col* <• Une 17 Vol. II. p. 41, line 97 Vol. III. p. 539, col. I, line 45 VoL I. p. 106, line i \ 99 P* 415* col. I, line ai y/or Giacomo read Giovanni VoL III. p. 558, coL a, line 95 j VoL I. p. 18a, line %\^ for Opot read^Opn Vol. III. p. 499. In the 6gnre» CV ihoald be a thick line. iTM'l EUCUD PREFACE \"nnHERE never has been, and till we see it we nev< ^ X shall believe that there can be, a system of geometi worthy of the name, which has any material departures (we d not speak of corrections or extensions or developments) froi the plan laid down by Euclid," De Morgan wrote thus i October 1848 (Short supplementary remarks on the first s\ Books of EuclicTs Elements in the Companion to the Almam for 1849); and I do not think that, if he had been livin to-day, he would have seen reason to revise the opinion s deliberately pronounced sixty years ago. It is true that in th interval much valuable work has been done on the continei in the investigation of the first principles, including th . formulation and classification of axioms or postulates whic are necessary to make good the deficiencies of Euclid's ow explicit postulates and axioms and to justify the furth< assumptions which he tacitly makes in certain proposition content apparently to let their truth be inferred from observ; tion of the figures as drawn ; but, once the first principles ai disposed of, the body of doctrine contained in the recent tex books of elementary geometry does not, and from the natui of the case cannot, show any substantial differences from th; set forth in the Elements. In England it would seem that f< less of scientific value has been done ; the efforts of a multitud of writers have rather been directed towards producing alte; natives for Euclid which shall be more suitable, that is to sa; easier, for schoolboys. It is of course not surprising that, i vi PREFACE these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute a *' royal road to geometry " ; the marvel is that a book v^^hich was not written for schoolboys but for grown men (as all internal evidence shows^ and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of " practical " geometry) should have held its own as a school- book for so long» And now that Euclid's proofs and arrange- ment are no longer required from candidates at examinations there has been a rush of competitors anxious to be first in the field with a new text-book on the more " practical " lines which now find so much favour. The natural desire of each teacher who writes such a text- book is to give prominence to some special nostrum which he has found successful with pupils. One result is, too often, a loss of a due sense of proportion ; and, in any case, it is inevitable that there should be great diversity of treatment. It was with reference to such a danger that Lardner wrote in 1846 : '* Euclid once superseded, every teacher would esteem his own work the best, and every school would have its own class book. All that rigour and exactitude which have so long excited the admiration of men of science would be at an end. These very words would lose all definite meaning. Every school would have a different standard ; matter of assumption in one being matter of demonstration in another; until, at length, Geometry, in the ancient sense of the word, would be altogether frittered away or be only considergd as a particular application of Arithmetic and Algebra/j It is, perhaps, too early yet to prophesy what will be the ultimate outcome of the new order of things ; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more. But the case for a new edition of Euclid is independent of any controversies as to how geometry shall be taught to schoolboys. Euclid's work will live long after all the text-books PREFACE (of the present day are superseded and forgotten. It is on of the noblest monuments of antiquity ; no mathematicia worthy of the name can afford not to know Euclid, the r« Euclid as distinct from any revised or rewritten version which will serve for schoolboys or engineers. And, to kno^ L Euclid, it is necessary to know his language, and, so far as ' can be traced, the history of the "elements" which h ; collected in his immortal work. \ This brings me to the raison d'Hre of the present editioi A new translation from the Greek was necessary for tw I reasons. First, though some time has elapsed since th '^ appearance of Heiberg's definitive text and prolegomens I published between 1883 and 1888, there has not been, so fa ^ as I know, any attempt to make a faithful translation from I into English even of the Books which are commonly reac ^ And, secondly, the other Books, vii. to x. and xiii., were nc ' included by Simson and the editors who followed him, c : apparently in any English translation since Williamson (178 1 — 8), so that they are now practically inaccessible t English readers in any form. In the matter of notes, the edition of the first six Book ' in Greek and Latin with notes by Camerer and Haubc (Berlin, 1824 — 5) is a perfect mine of information. It woul h have been practically impossible to make the notes mor \ exhaustive at the time when they were written. But th researches of the last thirty or forty years into the history < mathematics (I need only mention such names as those < r Bretschneider, Hankel, Moritz Cantor, Hultsch, Paul Tannery j^ Zeuthen, Loria, and Heiberg) have put the whole subjec i( upon a different plane. I have endeavoured in this editio to take account of all the main results of these researches u to the present date. Thus, so far as the geometrical Book ^ are concerned, my notes are intended to form a sort c (f dictionary of the history of elementary geometry, arrange according to subjects ; while the notes on the arithmetic* Books VII. — IX. and on Book x. follow the same plan. viii PREFACE I desire to express here my thanks to my brother, Dr R. S* Heath, Vice- Principal of Birmingham University, for suggestions on the proof sheets and, in particular p for the reference to the parallelism between Euclid's definition of proportion and Dedekind's theory of irrationals, to Mr R. D* Hicks for advice on a number of difficult points of translation, to Professor A. A. Bevan for help in the transliteration of Arabic names, and to the Curators and Librarian of the Bodleian Library for permission to reproduce, as frontispiece, a page from the famous Bodleian MS. of the Ehments^ Lastly, my best acknowledgments are due to the Syndics of the Cambridge University Press for their ready acceptance of the work, and for the zealous and efficient cooperation of their staff which has much lightened the labour of seeing the book through the Press, T. U H, NovembcTy 1908. X • • • r . ■.•••: • • • ^ INTRODUCTION. I CHAPTER I. EUCLID AND THE TRADITIONS ABOUT HIM. As in the case of the other great mathematicians of Greece, so in Euclid's case, we have only the most meagre particulars of the life and personality of the man^ Most of what we have is contained in the passage of Proclus' summary relating to him, which is as follows^: " Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collect- ing many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived* in Uie time of the first Ptolemy. For Archimedes, who came imme- diately after the first (Ptolemy)', makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry*. He is then younger than the pupils of Plato but older than Eratosthenes and Archimedes ; for the latter were contemporary with one another, as Eratosthenes some- where says." This passage shows that even Proclus had no direct knowledge ^ of Euclici s birthplace or of the date of his birth or death. He pro- - ceeds by inference. Since Archimedes lived just after the first I ^ Proclus, ed. Friedlein, p. 68, 6 — lo. ' * The word yfywe must apparently mean ''flourished," as Heiberg understands it ^ ^LitterargeschichtNche StudUn iiber Euklid, 1881, p. 16), not "was bom," as Hankel took it : otherwise part of Proclus' argument would lose its cogency. ^ 'So Heiberg understands /iri/SoXiiw r^ wptimfi (sc. nroXefuUy). Friedlein's text has I Kol between iwifioKunf and nf wptirtfi; and it is rignt to remark that another reading is U KolipTif wpilbrtf (without irtfiaXtitf) which has been translated " in his first ^>(," by wmch is understood On the Sphere and Cylmder i., where (i) in Prop- ^ ure the words *' let BC be made equal to D by the seamd (proposition) of the first of Euclid's (books)," and (a) in Prop. 6 the words " For these things are handed down in the Elements " (without the name of Euclid). Heiberg thinks the former passage is rdPerred to, and tnat Proclus must therefore have had before him the words " by the second of the first of Euclid ** : a fair proof that they are genuine, though in themselves &iey would be somewhat suspicious, f ^ The same story is told in Stobaeus, Eel, (11. p. 338, 30, ed. Wachsmuth) about Alexander and Menaechmus. Alexander is represented as having asked Menaechmiis to teach him geometry concisely, but he replied : '*0 kinff, through the country there are royal roads and roads for common citixens, but in geometry there is one road for all.*' H. E. "^ I t f y 4 t u/'", INTRODUCTION [gh j Ptolemy/'ftoS Archimedes mentions Euclid^ while there is an anecdote dhout'J^f^e Ptolemy and Euclid, therefore EucUd hVed in the time of the fii^l Ptolemy. .W^'may infer then from Proclus that Euclid was intermediate between the first pupils of Plato and Archimedes. Now Plato died in ■3^ J Archimedes lived 287-212, Eratosthenes 276-194 B.C Thus r, Ej/clid must have flourished c. 300 B,C, which date agrees well with V*the fact that Ptolemy reigned from 306 to 2 S3 B.C /•■ It is most probable that Euclid received his mathematical training * in Athens from the pupils of Plato; for most of the geometers who could have taught him were of that school, and it was in Athens that the older writers of elements, and the other mathematicians on whose works Euclid's Elemtnts depend, had lived and taught He may himself have been a Platonist, but this does not follow from the state- ments of Proclus on the subject. Proclus says namely that he was of the school of Plato and in close touch with that philosophy \ But this was only an attempt of a New Platonist to connect Euclid with his philosophy I as is clear from the next words in the same sentence, •* for which reason also he set before himself, as the end of the whole Elements, the construction of the so-called Platonic figures" It is evident that it was only an idea of Proclus* own to infer that Euclid was a Platonist because his Elemmts end with the investigation of the five regular solids, since a later passage shows him hard put to it to reconcile the view that the construction of the five regular solids was the end and aim of the Eiements with the obvious fact that they were intended to supply a foundation for the study of geometry in general, "to make perfect the understanding of the learner in regard - to the whole of geometry'" To get out of the difficulty he says' that, if one should ask him what was the aim (<rico7ro9) of the treatise, he would reply by making a distinction between Euclid's intentions (i) as regards the subjects with which his investigations are concerned, (2) as regards the learner, and would say as regards (I) that "the whole of the geometer's argument is concerned with the cosmic figures." This latter statement is obviously incorrect It is true that Euclid's Elements end with the construction of the five regular solids; but the plan i metrical portion has no direct relation to them, and the arithmetical no relation at all ; the propositions about them are merely the conclusion of the stereo metrical division of the work. One thing is however certain, namely that Euclid taught, and founded a school, at Alexandria, This is clear from the remark of Pappus about Apollonius*: *'he spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought** It is in the same passage that Pappus makes a remark which might J to an unwary reader, seem to throw some light on the ^ Proclus, p. ^1 90, Kol rf rpoatp^ct £j nXorbuvtr&f iart jral rj ^X«<ra^9 Tm:CfrQ alxiUitt. ■ (W^, p* 71, 8. * ibid. p» 70, 19 ^qq, * PttppU^, VII. p* ISjSt 10— iIt inAT^oXd^at r«r itwh £l^j(X«iJcM^ pud^itmlt ir 'AXtfardpeffi I CH. i] EUCLID AND TRADITIONS ABOUT HIM personality of Euclid. He is speaking about Apollonius' prefac to the first book of his Conies^ where he says that Euclid had no I completely worked out the synthesis of the "three- and four-Iin< r* locus," which in fact was not possible without some theorems firs discovered by himself. Pappus says on this^: "Now Euclid- "^ regarding Aristaeus as deserving credit for the discoveries he ha( already made in conies, and without anticipating him or wishing t< construct ane\v the same system (such was his scrupulous fairness an( V his exemplary kindliness towards all who could advance mathematica ' science to however small an extent), being moreover in no wise con tentious and, though exact, yet no braggart like the other [ApoUonius . — ^wrote so much about the locus as was possible by means of th< • conies of Aristaeus, without claiming completeness for his demonstra tions." It is however evident, when the passage is examined in it context, that Pappus is not following any tradition in giving thi ^ account of Euclid: he was offended by the terms of Apolloniuj ' reference to Euclid, which seemed to him unjust, and he drew \ \ fancy picture of Euclid in order to show ApoUonius in a relativel] ^ unfavourable light Another story is told of Euclid which one would like to believe true I According to^tobaeus*, " some one who had begun to read geometr with EuclidT^en he had learnt the first theorem, asked Euclid, ' Bu f what shall 1 get by learning these things ? ' Euclid called his slav \ and said ' Give him threepence, since he must make gain out of wha j he learns.'" i In the middle ages most translators and editors spoke of Eucli( as Euclid of Megara, This description arose out of a confusioi between our Euclid and the philosopher Euclid of Megara who livec about 400 B.C The first trace of this confusion appears in Valeriu Maximus (in the time of Tiberius) who says' that Plato, on beinj appealed to for a solution of the problem of doubling the cubica altar, sent the inquirers to "Euclid the geometer." There is no doub about the reading, although an early commentator on Valeriu Maximus wanted to correct "Eucliden" into **EudoxufnP and thi correction is clearly right. But, if Valerius Maximus took Euclid th( geometer for a contemporary of Plato, it could only be througl confusing him with Euclid of Megara. The first specific reference t Euclid as Euclid of Megara belongs to the 14th century, occurring i: the virofunjfiarur/jLoi of Theodorus Metochita (d. 1332) who speaks c " Euclid of Megara, the Socratic philosopher, contemporary of Plato, as the author of treatises on plane and solid geometry, data, optic etc. : and a Paris MS. of the 14th century has " Euclidis philosoph Socratici liber elementorum.'' The misunderstanding was genera in the period from Campanus' translation (Venice 1482) to those c Tartaglia (Venice 1565) and Candalla (Paris 1566). But on Constantinus Lascaris (d. about 1493) had already made the prope ^ Pappus, VII. pp. 676, 45 — 678, 6. Hultsch, it is true, brackets the whole possag pp. 676, 95—^78, 15, but apparently on the ground of the diction only. * Stobaeus, /,e. * viii. la, ext. i. ^m 4 INTRODUCTION [cH. i distinction by saying of our Euclid that " he was different from him of Megara of whom Laertius wrote, and who wrote dialogues*'' ; and to Commandinus belongs the credit of being the first translator' to put the matter beyond doubt : *' Let us then free a number of people from the error by which they have been induced to believe that our Euclid is the same as the philosopher of Megara" etc Another idea, that Euclid was born at Gela in Sicily, is due to the same confusion, being based on Diogenes Laertius' description' of the philosopher Euclid as being *'of Megara, or, according to some, of Gela, as Alexander says in the ^ea^o^^atV In view of the poverty of Greek tradition on the subject even as early as the time of Proclus (410-485 a.d,), we must necessarily take a*m grano the apparently circumstantial accounts of Euclid given by Arabian authors ; and indeed the origin of their stories can b^ explained as the result (i) of the Arabian tendency to romancep and (2) of misunderstandings. We read* that " Euclid, son of Naucrates, grandson of Zenarchus* called the author of geometry, a philosopher of somewhat ancient date, a Greek by nationality domiciled at Damascus, bom at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled the foundation or elements of geometry, a subject in which no more general treatise existed before among the Greeks :fnay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. Hence ako Greek, Roman and Arabian geometers not a few, who undertook the task of illustrating this work, published commentaries, scholia, and notes upon it, and made an abridgment of the work itself For this reason the Greek philosophers used to post up on the doors of their schools the weli*known notice : * Let no one come to our school, who has not first learned the elements of Euclid.*"! The details at the beginning of this extract cannot be derived from Greek sources, for even Proclus did not know anything about Euclid's father, while it was not the Greek habit to record the names of grandfathers, as the Arabians commonly did. Damascus and Tyre were no doubt brought in to gratify a desire which the Arabians always showed to connect famous Greeks in some way or other with the East Thus Naslraddlnj the translator of the Elemenis^ who was of Tus in Khurasan, actually makes Euclid out to have been "Thusinus" also* The readiness of the Arabians to run away with an idea is illustrated by the last words ^ Letter to Fe maud us Acuna^ printed in Mauroljcus, HUiima SuiHa^f foL 91 r> (see Heiberg, Evklid-Studien^ pp. ^i — 3, 45), ' Prcf»ce to translation (risauri, 'S?^)* * Diog. L. II. io6t p* 58 ed. Cobet. < Casiri, BiMiptkeca Arabka^Hispema SicuriaUmiiy I. p. 339* CasiH's source i» *1- Qlftl (d- 1343), the author of the Tt^rikh al-HukamSy a collection of biographies of phi- losophers, mathematicians, a^tronotncrs eta * The Fihrist says *'son of Naucratei, the son of Berenice (?) " {set Suter's translatLOO in Ahkandlungm ittr Gts<:h. d. Math. VI. Heft, 1893, p, 16 J, ' The same predilection made the Ambs de^ritie Pythagoras as a P^P^^ ^^ ^^^ ^^ Salome, Hipmrchus as the exponent of Chaldaean philosophy or as ibe Chaldaean, Archi- medes as an Egyptian etc. (H&ji Khalfaj Lexicon Sibiic^apkuum, and Casiri). CH. i] EUCLID AND TRADITIONS ABOUT HIM 5 of the extract Everyone knows the story of Plato's inscription over the porch of the Academy : " let no one unversed in geometry enter my doors " ; the Arab turned geometry into Euclid s geometry, and told the story of Greek philosophers in general and 'Hheir Academies." Equally remarkable are the Arabian accounts of the relation of Euclid and Apollonius\ According to them the Elements were originally written, not by Euclid, but by a man whose name was Apollonius, a carpenter, who wrote the work in 15 books or sections*. In the course of time some of the work was lost and the rest became disarranged, so that one of the kings at Alexandria who desired to study geometry and to master this treatise in particular first questioned about it certain learned men who visited him and then sent for Euclid who was at that time famous as a geometer, and asked him to revise and complete the work and reduce it to order. Euclid then re-wrote it in 13 books which were thereafter known by his name. (According to another version Euclid composed the 13 books out of commentaries which he had published on two books of Apollonius on conies and out of introductory matter added to the doctrine of the five regular solids.) To the thirteen books were added two more books, the work of others (though some attribute these also to Euclid) which contain several things not mentioned by Apollonius. According to another version Hypsicles, a pupil of Euclid at Alexandria, offered to the king and published Books xiv. and xv., it being also stated that Hypsicles had "discovered" the books, by which it appears to be suggested that Hypsicles had edited them from materials left by Euclid. We observe here the correct statement that Books XI v. and XV. were not written by Euclid, but along with it the incorrect informa- tion that Hypsicles, the author of Book xiv., wrote Book xv. also. The whole of the fable about Apollonius having preceded Euclid and having written the Elements appears to have been evolved out of the preface to Book xiv. by Hypsicles, and in this way ; the Book must in early times have been attributed to Euclid, and the inference based upon this assumption was left uncorrected afterwards when it was recognised that Hypsicles was the author. The preface is worth quoting : " Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of their common interest in mathematics. And once, when ^ The authorities for these statements (quoted by Casiri and Hajl Khalfa are al-KindI*s tract di imtituto Uhri Euclidis (al-Kindl died about 87^) and a commentary by (^ftd1z3de ar-RumI (d. about '440) on a book called Ashkdl at-ta: sis (fundamental propositions) by Ashraf Shamsaddin as-Samarqandl (^. 1176) consisting of elucidations of 35 propositions selected from the first books of Euclid. Na^Iraddln likewise says that Euclid cut out two of 15 books of elements then existing and published the rest under his own name. According to Q&dlz&de the king heard that there was a celebrated geometer named Euclid at Tyre: Naflr- adciln says that he sent for Euclid of Tus. * So says the Fihrist, Suter (op! cit, p. 49) thinks that the author of the Fihrist did not suppose Apollonius of Perga to be the writer of the EUtnents^ as later Arabian authorities did, but that he distinguished another Apollonius whom he odls **a carpenter." Sutor's argument is based on the fact that the Fihrisfs article on Apollonius (of Perga) says nothins; of the Elements^ and that it gives the three great mathematicians, Euclid, Archimedes and Apollonius, in the correct chronological order. K A INTRODUCTION [cH. i examining the treatise written by Apollonius about the comparison between the dodecahedron and the icosahedron inscribed in the same sphere, (showing) what ratio they have to one another, they thought that Apollonius had not expounded this matter properly, and accordingly they emended the exposition, as I was able to learn from my father. And I myself, later, fell in with another book published by Apollonius, containing a demonstration relating to the subject, and I was greatly interested in the investigation of the problem. The book published by Apollonius is accessible to all— for it has a large circulation, having apparently been carefully written out later — ^but I decided to send you the comments which seem to me to be necessary, for you will through your proficiency in mathe- matics in general and in geometry in particular form an expert judgment on what I am about to say, and you will lend a kindly ear to my disquisition for the sake of your friendship to my father and your goodwill to me." The idea that Apollonius preceded Euclid must evidently have been derived from the passage just quoted. It explains other things besides, Basilides must have been confused with ffatrtX^vs, and we have a probable explanation of the "Alexandrian king,'* and of the "learned men who visited'* Alexandria. It is possible also that in the '* Tyrian '' of Hypsicles' preface we have the origin of the notion that Euclid was bom in Tyre, These inferences argue, no doubt, very defective knowledge of Greek : but we could expect no better from those who took the Organon of Aristotle to be " instrumentum musicum pneumaticum/' and who explained the name of Euclid, which they variously pronounced as Uclid^s or Icludes, to be com- pounded of Ucli a key, and Dis a measure, or, as some say, geometry, so that Udides is equivalent to the k^ of geametfyl Lastly the alternative version, given in brackets above, which says that Euclid made the Eiimenis out of commentaries which he wrote on two books of Apollonius on conies and prolegomena added to the doctrine of the five solids, seems to have arisen, through a like confusion, out of a later passage' in Hypsicles* Book xiv, : "And this is expounded by Aristaeus in the book entitled 'Comparison of the five figures,' and by Apollonius in the second edition of his comparison of the dodecahedron with the icosahedron.'* The "doctrine of the five solids " in the Arabic must be the " Comparison of the five figures " in the passage of Hypsicles, for nowhere else have we any information about a work bearing this title, nor can the Arabians have had. The reference to the tivo books of Apollonius on conks will then be the result of mixing up the fact that Apollonius wrote a book on conies with the second edition of the other work mentioned by Hypsicles. We do not find elsewhere in Arabian authors any mention of a commentary by Euclid on Apollonius and Aristaeus : so that the story in the passage quoted is really no more than a variation of the fable that the Elements were the work of Apollonius, ^ Heiberg's Eaclld, toL v* p. 6, f CHAPTER II. y EUCLID'S OTHER WORKS. In giving a list of the Euclidean treatises other than the EUmeHts, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematics ^ I will take first the works which are mentioned by Greek authors. I. The Pseudaria, I mention this first because Proclus refers to it in the general remarks in praise of the Elements which he gives immediately after the mention of Euclid in his summary. He says': "But, inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms, and to avoid being misled. This treatise, by which he puts tiiis machinery in Our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry." The book is considered to be irreparably lost We may conclude however from the connexion of it with the Elenunts and the reference to its usefulness for beginners that it did not go outside the domain of elementary geometry*. ^ Heibeig gives very exhaustive details in his LitterargeschichtlUhe Studien iiber Euklid\ the best of the shorter accounts are those of Cantor (Gesch, d. Afaih. i,, 1907, pp. 378 — 394) and Loria (II periodo aureo delta geometria greca^ p. 9 and pp. 63 — 85). • Proclus, p. 70, I— 18. ' Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his commentary on Aristotle's Sophistici EUnchi (fol. 15 bV. **Not only those (AeTxoc) which do not start from the principles of the science, under which the problem is classed... but also those which do start from the proper principles of the science but in some respect admit a paralogism, e.g. the Fseudographanata of Euclid." Tannery (BuU. des sciences math, et astr, 1* S^e, vi., 1883, i^ Partie, p. 147) conjectures that it may be from this treatise that the same commentator got his information about the quadratures of the circle by Antiphon and INTRODUCTION [cH. ii r 2. The Data The Data {Z€topi,iifa) are included by Pappus in the Treasury af Analysis {tqitok ava\v6^^o^\ and he describes their contents*. They are still concerned with elementary geometry, though forming part of the introduction to higher analysis. Their form is that of pro- positions proving that, if certain things in a figure are given (in magnitude, in species, etc.)* something else is given. The subject* matter Is much the same as that of the plani metrical books of the \ Elements, to which the Data are often supplementary. We shall see this later when we come to compare the propositions in the Elements which give us the means of solving the general quadratic equation with the corresponding propositions of the Data which give the solution. The Data may in fact be regarded as elementary exercises in analysis. It is not necessary to go more closely into the contents, as we have the full Greek text and the commentary by Marinus newly edited by Menge and therefore easily accessible* 3- The book On divisions {ofjignres). This work {ir^pl Biaif>icr€wv fit^iov) is mentioned by Proclus*. In one place he is speaking of the conception or definition {X0709) o{ figure, and of the divisibility of a figure into others differing from it in kind ; and he adds: "For the circle is divisible into parts unlike in definition or notion {dvofiGta t^ >Jrf^\ and so is each of the rectilineal figures ; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike**" "Like" and "unlike" here mean, not "similar*' and "dissimilar** in the technical sense, but "like" or "unlike in definition or notion'' (Xo^^): thus to divide a triangle into triangles would be to divide it into "like" figures, to divide a triangle into a triangle and a quadrilateral would be to divide it into "unlike" figures. The treatise is lost in Greek but has been discovered in the Arabic. First John Dee discovered a treatise De divisionibus by one Muhammad Bagdad] n us ^ and handed over a copy of it (in Latin) in 1563 to Commandinus, who published it, in Dee's name and his own, in IS70*, It was formerly supposed that Dee had himself translated BrysoD, to say aothing of the lunules of Hippocrates* I tliink however that there b an objection to this theory so far as regank Brysooi for Alexander distinctly lays that Biyson^s quadrature did not start from the proper priijcipJes of geometfy, hut frofn some priiuriples more generat. " Pappus, vn. p. 638. ' VoL VI ♦ in the Teiibner edition of EuctMii cftra otnnm by Heiberg and Menge. A translation of the Puta is also included In Simson a EucHd (though culturally his text left much to be desired)* ■ Proclusj p. 69 J 4. * ihid. J 44, 21— 16- * Steinschneider places him in the totlic. H. Sv^itt {BthH^ktca Maihtmaika, iVj, 1903, pp. 34, 17) ideoti^es him with Abu(Bekr) Moh. b. "Abdalbiql al-Ba^adl, QSdl 0ud^ey of M&rbtiln {drca lojo-n+t), to whom he also attributes the Libtrj^dti (? judicis) super dtamu^ Euelidii translated by Gherard of Cremona. * Df superfHtrum ditnsionihus liher Math&mdo Bagitadina aditriptui^ nurv primttm Uannis Dtt Londinensii it Fed^^i C^mfrmttdini Urhinntis eptra itt iu€fm tiMhiSi Plsaurt, 1570, afterwania bduded in Gregory's Euclid {Oxford, 1703)- CH. ii] EUCLID'S OTHER WORKS 9 the tract into Latin from the Arabic*; but it now appears certain' that he found it in Latin in a Cotton MS. now in the British Museum. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible MS. had caused him much trouble and (in a later passage) speaks of ••the actual, very ancient, copy from which I wrote out..** (in ipso unde descripsi vetustissimo exemplari). The Latin translation of this tract from the Arabic was probably made by Gherard of Cremona (1114-1187), among the list of whose numerous translations a "liber divisionum" occurs. The Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adapta- tion of it ; it contains mistakes and unmathematical expressions, and moreover does not contain the propositions about the division of a circle alluded to by Proclus. Hence it can scarcely have contained more than a fragment of Euclid's work. \ But Woepcke found in a MS. af Paris a treatise in Arabic on the division of figures, which he translated and published in 1851*. It is expressly attributed to Euclid in the MS. and corresponds to the description of it by Proclus. Generally speaking, the divisions are divisions into figures of the same kind as the original figures, e.g. of triangles into triangles; but there are also divisions into " unlike*' figures, e.g. that of a triangle by a straight line parallel to the base. The missing propositions about the division of a circle are also here : **to divide into two equal parts a given figure bounded by an arc of a circle and two straight lines including a given angle" and "to draw in a given circle two parallel straight lines cutting off a certain part of the circle." Unfortunately the proofs are given of only four propositions (including the two last mentioned) out of 36, because the Arabic translator found them too easy and omitted them. To illustrate the character of the problems dealt with I need only take one more example : " To cut off a certain fraction from a (parallel-) trapezium by a straight line which passes through a given point lying inside pr outside the trapezium but so that a straight line can be drawn through it cutting both the parallel sides of the trapezium." The genuineness of the treatise edited by Woepcke is attested by the facts that the four proofs which remain are elegant and depend on propositions in the Elements, and that there is a lemma with a true Greek ring: "to apply to a straight line a rectangle equal to the rectangle contained hy AB, AC and deficient by a square'' Moreover the treatise is no fragment, but finishes with the words " end of the treatise," and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is .not only Euclid's own work but the whole of it*. A restoration of the work, with proofs, was attempted ^ Heiberg, Euklid'Studien, p. 13. ' H. Suter in Bibliotkeca MaihematUa, iv,, 1905-6, pp. 331—3. • Journal Asiatique, 185 1, p. 133 sgq. * We are told by Casiri that Th&bit b. Qarra emended the translation of the Uber de divisumibus\ but Ofterdinger seems to be wrong in sajring that according to Gartz (/?/ i$Uer- prUUms et expianaioribus EuciuUs AraHcis schediasma histancum, HaUe, 1813) there is a 10 INTRODUCTION [ch, h by OfterdingerS who however does not give Woepckc's props. 30, 31, 34* 35i 3^^ 4. The Porisms. It is not possible to give in this place any account of the con- troversies about the contents and signiEcance of the three lost books of Porisms, or of the important attempts by Robert Simson and Chasles to restore the work. These may be said to form a whole literature, references to which will be found most abundantly given by Heiberg and Loria, the former of whom has treated the subject from the philological point of view; most exhaustively, while the latter, founding himself generally on Heiberg, has added useful detailsj from the mathematical side, relating to the attempted restora- tions, etc,' It must suffice here to give an extract from the only original source of information about the nature and contents of the P^nsms, namely Pappus', In his general preface about the books composing the Treasury of Afmfysis (totto? aWXvd^evo^) he says : "After the Tangencies (of Apollonius) come, in three books, the Porisms of Euclid, [in the view of many] a collection most ingeniously devised for the analysis of the more weighty problems, [and] although nature presents an unlimited number of such porisms', [they have added nothing to what was written originally by Euclid, except that some before my time have shown their want of taste by adding to a few (of the propositions) second proofs^ each (proposition) admitting of a definite number of demonstrations, as we have shown, and Euclid having given one for each, namely that which is the most lucid* These porlsms embody a theory subtle, natural, necessary, and of considerable generality, which is fascinating to those >vho can see and produce results]. ** Now all the varieties of porisms belong, neither to theorems nor problems, but to a species occupying a sort of intermediate position [so that their enunciations can be formed like those of either theorems or problems], the result being that, of the great number of geometers, some regarded them as of the class of theorems, and others of pro- blems, looking only to the form of the proposition. But that the ancients knew better the difference between these three things, is clear from the 'definitions* For they said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed J a problem that which is thrown out with a view to the construction of the very thing proposed, and a porism that which is proposed with a view to the producing of the very thing proposed. [But this definition of the porism was changed by the more recent writers who could not produce everything, but used these elements CDnapkt« MS. of Tfaabit'9 tmnslAtioQ m ihe Escurml. I cannot find any such staCemcDt in Gaitz. ^ L. F. Oftcvdingeri Stiirc^ tur WUdfrkersi£liung tUr Sckrifl dei Euklides iiber dk Tkdilung d^ Figurm^ Ulm, 185^- ' Heiberg, Eukiid-S(udien^ pp, 56^ — 79, uwl Loha, H pftiodo tmrea ddla geometric gt^tif pp. 70— Si, 13J— 5. ' pAppO£, cd. HuJudi, VII, pp, 648—660. I put in square brackeu the words bracket^ bj HuJtsck * I adopt Heibefg'a reading of a comma here instead of a full stop* — ^ * CH. n] EUCLID'S OTHER WORKS ii and proved only the fact that that which is sought really exists, but did not produce it^ and were accordingly confuted by the definition and the whole doctrine. They based their definition on an incidental characteristic, thus : A porism is that which falls short of a locus- theorem in respect of its hypothesis*. Of this kind of porisms loci are a species, and they abound in the Treasury of Analysis ; but this species has been collected, named and handed down separately from the porisms, because it is more widely diffused than the other species]. But it has further become characteristic of porisms that, owing to their complication, the enunciations are put in a contracted form, much being by usage left to be understood; so that many geometers understand them only in a partial way and are ignorant of the more essential features of their contents. ^[Now to comprehend a number of propositions in one enunciation is by no means easy in these porisms, because Euclid himself has not in fact given many of each species, but chosen, for examples, one or a few out of a great multitude^ But at the beginning of the first book he has given some propositions, to the number of ten, of one species, namely that more fruitful species consisting of loci.] Consequently, finding that these admitted of being comprehended in one enunciation, we have set it out thus: If, in a system of four straight lines^ which cut each other two and two, three points on one straight line be given while the rest except one lie on different straight lines given in position, the remaining point also will lie on a straight line given in position'. ^ Heiberg points out that Props. 5—^ of Archimedes' treatise On Sfirals are porisms in this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K, It is then pK)6sible, says Archimedes, to araw a straight line q g KHF^ meeting the circumference in H and the tangent in /*, such that FH\HK<(%szBiri\c, where c is the circomference of any circle. To prove this he assumes the following construction. E being any straight line greater than r, he says : let KG be parallel to DF^ "and let the line GH equal to i? be placed verging to the point B.^^ Archimedes must of course nave known how to effect this construction, which requires conies. But that it is possible requires very little argument, for if we draw any straight line BHG meeting the circle in ZTand KG in G^ it is obvious that as G moves away from C, HG becomes neater and greater and may be made as great as we please. The '* later writers *' would no doubt have contented themselves with this considera- tion without actually constructing HG, ' As Heibere says, this translation is made certain by a preceding passage of Pappus 0>> 648, I — 3) where he compares two enunciations, the latter of which *' falls short of the former in hypothesis but goes oeyond it in requirement,^'* E.g. the first enunciation requiring us, given three circles, to draw a circle touching all three, the second may require us, given only two circles (one less datum), to draw a circle touching them and of a given siie (an extra requirement). * I translate Heiberg's reading with a full stop here followed by rpbt dffxv ^ ^h*"^ l""/^ ^}dt^ {ifiofjJpop) Hultsch] Tov vpibrou fiipxiov..., * The four straight lines are described in the text as (the sides) hrrlov 4 wapvwrlov, Le. sides of two sorts of quadrilaterals which Simson tries to explain (see p. no of the Index GraecitcUis of Hultsch*s edition of Pappus). * In other words (Cbasles, p. «3 ; Loria, p. 73) if a triangle be so deformed that each of its sides turns about one of three points in a straight line, and two of its vertices lie on two straight lines given in position, the third vertex will klso lie on a straight line. tJ INTRODUCTION [ch. ii **Thts has only been enunciated of four straight line*!, of which not more than two pass through the same point, but it is not known (to most people) that it is true of any assigned number of straight lines if enunciated thus: If any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, and if each of those which are on another (of them) lie on a straight line given in position — or still more generally thus : if any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, while of the other points of intersection in multitude equal to a triangular number a number corresponding to the side of this triangular number tie respectively on straight lines given in position, provided that of these latter points no three are at the angular points of a triangle {sc. having for sides three of the given straight lines)— each of the remaining points will lie on a straight h'ne given in position*. " It is probable that the writer of the Elements was not unaware of this but that he only set out the principle ; and he seems, in the case of all the porisms, to have laid down the principles and the seed only [of many important things], the kinds of which should be distinguished according to the differences, not of their hypotheses, but of the results and the things sought [All the hypotheses are different from one another because they are entirely special, but each of the results and things sought, being one and the same, follow from many different hypotheses.] "We must then in the first book distinguish the following kinds of things sought : "At the beginning of the book* is this proposition : I, ' If from two givm points straight lines be drawn meeting on a straight line given in position^ and one cut off from a straight Urn given in position (a segment measured) to a given point on i/, the other wilt also cut off from afwther {straight line a segment) having to the first a given ratio' *■ Following on this (we have to prove) IL that such and such a point lies on a straight line given in position ; II L that the ratio of such and such a pair of straight lines is given ; '* etc. etc. (up to xxix.). ^The three books of the porisms contain 38 lemmas; of the theorems themselves there are 171." ^ Loria (p. 7^ note) gives the meaning of this as follows, pointing out that Simson was the discoverer o( it : " Iia complete M-lateral be deformed so that its sides respectively turn about n points on a straight line, and (it- i) of its n (n- i)/a vertices move on as many straight Imes, the other (m- i)(ii-9)/a of its vertices likewise move on as many straight lines: but it is necessary that it shoula be impossible to form with the (m- i) vertices any triangle having for sides the sides of the polygon." ' Reading, with Keiberg, roH fitfiXlov [roC r Hultsch]. \ CH. ii] EUCUiyS OTHER WORKS 13 Pappus further gives lemmas to the Porisms (pp. 866 — 918, ed, Hultsch). With Pappus' account of Porisms must be compared the passages of Proclus on the same subject Proclus distinguishes two senses in which the word ir6pur/jui is used. The first is that of corollary where something appears as an incidental result of a proposition, obtained without trouble or special seeking, a sort of bonus which the investi- gation has presented us with^ The other sense is that of Euclid's Porisf9ts\ In this sense' ^^porism is the name given to things which are sought, but need some finding and are neither pure bringing into existence nor simple theoretic argument For (to prove) that the angles at the base of isosceles triangles are equal is a matter of theoretic argument, and it is with reference to things existing that such knowl^ge is (obtained). But to bisect an angle, to construct a triangle, to cut off, or to place — all these things demand the making of something ; and to find the centre of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or the like, is in some sort between theorems and problems. For in these cases there is no bringing into existence of the things sought, but finding of them, nor is the procedure purely theoretic. For it is necessary to bring that which is sought into view and exhibit it to the eye. Such are the porisms which Euclid wrote, and arranged in three books of Porisms. Proclus' definition thus agrees well enough with the first, ** older," definition of Pappus. A porism occupies a place between a theorem and a problem: it deals with something dXreaidy existing, ^s a theorem does, but has to find it (e.g. the centre of a circle), and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce some- thing not previously existing. Thus, besides IIL i of the Elements and X. 3, 4 mentioned by Proclus, the following propositions are real porisms: ill. 25, VL 11— 13, vii. 33, 34, 36, 39, vill. 2,4, x. 10, XIIL 18. Similarly in Archimedes On the Sphere and Cylifuler I. 2 — 6 might be called porisms. The enunciation given by Pappus as comprehending ten of Euclid's propositions may not reproduce the form of Euclid's enunciations ; but, comparing the result to be proved, that certain points lie on straight lines given in position, with the class indicated by ll. above, where the question is of such and such a point lying on a straight line given in position, and with other classes, e.g. (v.) that such and such a line is given in position, (vi.) that such and such a line verges to a given point, (XXVII.) that there exists a given point such that straight lines drawn from it to such and such (circles) will contain a triangle given in species, we may conclude that a usual form of a porism was ^* to prove that it is possible to find a point with such and such a property" * Proclus, pp. a I a, 14 ; 301, «i. ' ibid. p. 313, I a. "The term porism is used of certain problems, like the Porisms written by Euclid." * ihid, pp. 301, 35 sqq. 14 INTRODUCTION [CB. ii or "a straight line on which lie all the points satisfying given conditions" etc. Simson defined a porism thus : " Porisma est propositio iti qua proponitur dem oust rare rem aliquam, vel pi u res datas esse^ cuij vel quibus, ut et cuilibet ex rebus innumeriSj non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, con venire ostendendum est affecttonem quandam communem in propositione descriptamV From the above it is easy to understand Pappus' statement that loci constitute a large class of porisms. A hcus is well defined by Simson thus : '* Locus est propositio in qua propositum est datam esse demonstrare, vel in venire lineam aut superficiem cuius quodlibet punctum, vel superficiem in qua quaelibet linea data lege descripta, communem quandam habet proprietatem in propositione descriptam." Heiberg cites an excellent instance of a l<fats which is Sip&rism, namely the following proposition quoted by Eutocius* from the P/afie Zati of ApoHonius : *' Given two points in a plane, and a ratio between unequal straight lines, it is possible to draw, in the plane, a circle such that the straight lines drawn from the given points to meet on the circumference of the circle have (to one another) a ratio the same as the given ratio/* A difficult point, however, arises on the passage of Pappus, which says that a porism is "that which, in respect of its hypothesis, falls short of a locus-theorem" (roTrueov ^€6>/j?J^aTo?)- Heiberg explains it by comparing the porism from ApoHonius' Platit Xa^i just given with Pappus' enunciation of the same thing, to the effect that, if from two given points two straight lines be drawn meeting in a point, and these straight lines have to one another a given ratio, the point will lie on either a straight line or a circumference of a circle given in position. Heiberg observes that in this latter enunciation something is taken into the hypothesis which was not in the hypothesis of the enunciation of the porism, viz. "that the ratio of the straight lines is the same." I confess this does not seem to me satisfactory : for there is no real difference between the enunciations, and the supposed difference in hypothesis is very like playing with words. Chasles says : " Ct qui const itu€ U porisme est ct qui manque d I'hypoth^e d'un tASreme local (en d 'aut res termes, le porisme est infifrieur, par Vhypoth^se, au th^orimc local; c'est-i*dire que quand quelques parties d'une pro- position locale n'ont pas dans T^nonc^ la determination qui leur est propre, cette proposition cesse d'etre regardee com me un th^rfeme et devient un porisme)/* But the subject still seems to require further elucidation. While there is so much that is obscure, it seems certain (i) that the Porisms were distinctly pait of higher geometry and not of elementary ^ This was thus expressed bj Cbasles i '* Le porisme ^ udc propositioti dons laquellc on dcmimde de d^montrer qu'unc chose ou plusieurs choses sont d^nn/es, qui, flinsi que iWe quelcouque d'une in^nUe d^autres choses non donates, mais dont chacune est avec des choses doDufes daoa u&e m^me relation, one une cert^ine propri^t^ comoiune, d^rite dam Ja pio^ position, " ■ Commentary on ApoUonius* C^ni^s (vol. ii. p, iSo^ ed. Heiberg). 1 I* CH. II] EUCLID'S OTHER WORKS 15 geometry, (2) that they contained propositions belonging to the modem theory of transversals and to projective geometry. It should i be remembered too that it was in the course of his researches on this subject that Chasles was led to the idea of anharmanic ratios. Lastly, allusion should be made to the theory of Zeuthen^ on the subject of the porisms. He observes that the only porism of which Pappus gives the complete enunciation, *' If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) towards a given point on it, the other will also cut off from another (straight line a segment) bearing to the first a given ratio," is also true if there be substituted for the first given straight line a conic regarded as the '' locus with respect to four lines," and that this extended porism can be used for completing Apollonius' exposition of that locus. Zeuthen concludes that the Porisms were in part by- products of the theory of conies and in part auxiliary means for the study of conies, and that Euclid called them by the same name as that applied to corollaries because they were corollaries with respect to conies. But there appears to be no evidence to confirm this conjecture. 5. The Surface-loci (tmto* irpo^ em<f>aif€ia). The two books on this subject are mentioned by Pappus as part of the Treasury of Analysis^. As the other works in the list which were on plane subjects dealt only with straight lines, circles, and conic sections, it is a priori likely that among the loci in this treatise (loci which are surfaces) were included such loci as were cones, cylinders and spheres. Beyond this all is conjecture based on two lemmas given by Pappus in connexion with the treatise. (i) The first of these lemmas' and the figure attached to it are not satisfactory as they stand, but a possible restoration is indicated by Tannery*. If the latter is right, it suggests that one of the loci contained all the points on the elliptical parallel sections of a cylinder and was therefore an oblique circular cylinder. Other assumptions with r^ard to the conditions to which the lines in the. figure may be subject would suggest that other loci dealt with were cones regarded as containing all points on particular elliptical parallel sections of the cones*. (2) In the second lemma Pappus states and gives a complete proof of the focus-and-directrix property of a conic, viz. that the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed line is a conic section^ which is an ellipse ^ a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unity\ Two conjectures are possible as to the application of this theorem in Euclid's Surface-loci, (a) It may have been used to prove that the locus of a point whose distance from a given straight ^ Die Lehre von den Ktgdschnitten im AUertum^ chapter viii. • Pappus, VII. p. 636. • ibid. vii. p. 1004. * Bulletin des sciences math, ei astron.., 3* S^rie, vi. 149. • Further particulars will be found in The IVarks e/ Archimedes, pp. Ixii — Ixiv, and in Zeuthen, Die Lehre von den JCegeischnitten^ p. 415 sqq. * Pappus, VII. pp. 1006 — 1014, and Hultsch*s Applendix, pp. 1370 — 3. i6 INTRODUCTION - [CH. ii line is in a given ratio to its distance from a given plane is a certain cone. {6} It may have been used to prove that the locus of a point whose distance from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axis^ Thus Chasles may have been correct in his conjecture that the Surfaa-ioci dealt with surfaces of revolution of the second degree and sections of the same^ 6, The Conks. Pappus says of this lost work; **The four books of Euclid's Conies were completed by ApoUonius, who added four more and gave us eight books of Conies'.*' It is probable that Euclid's work was lost even by Pappus' time, for he goes on to speak of '* Aristaeus, who wrote the still extant five books of Solid Loci connected with the conies/' Speaking of the relation of Euclid's work to that of Aristaeus on conies regarded as loci ^ Pappus says in a later passage (bracketed however by Hultsch) that Euclid, regarding Aristaeus as deserving credit for the discoveries he had already made in conies, did not (try to) anticipate him or construct anew the same system. We may no doubt conclude that the book by Aristaeus on solid loci preceded Euclid's on conies and was, at least in point of originality, more important Though both treatises dealt with the same subject-matter^ the object and the point of view were different ; had they been the same, Euclid could scarcely have refrained, as Pappus says he did, from attempting to improve upon the earlier treatise* No doubt Euclid wrote on the general theory of conies as Apollonius did, but confined himself to those properties which were necessary for the analysis of the Solid Loci of Aristaeus. The Conies of Euclid were evidently superseded by the treatise of Apollonius, As regards the contents of Euclid's Conies^ the most important source of our information is Archimedes, who frequently refers to propositions in conies as well known and not needing proof, adding in three cases that they are proved in the " elements of conies '* or in '*the conies," which expressions must clearly refer to the works of Aristaeus and Euclid** Euclid still used the old names for the conies (sections of a right- angled» acute*angled, or obtuse -angled cone), but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane parallel to the base (assuming the section to lie wholly between the apex of the cone and its base) and also by cutting a cylinder. This is expressly stated in a passage from the Phaenomma of Euclid about to te mentioned*, 7* The Phaenonuna. This is an astronomical work and is still extant A much inter- * For further d^taib sec Tlis Worki of Attkimidis, pp, bdv, latv, aod Zeuthen, /♦ f» * A^fH Misi^riyuff pp, 373—4' * Pappus, vri* p. 671. * For details of these proposition* see my Apoiknims 0/ Ptrga^ pp. stxiv, xnjtvi. ' See Heitjerg, Mukiid*StudUn^ p. 88. " If a cone or a cylinder be cut by a pkiae not parallel to the base» the section b a section of an acute -wigled conc^ which is like a shield '! CH.n] EUCLID'S OTHER WORKS 17 '( polated version appears in Gregor/s Euclid, and a much earlier and • f better recension is, says Heiberg\ contained in the MS. Vindobonensis M philos. Gr. 103, though the end of the treatise, from the middle of I prop. 16 to the last (18), is missing. The book consists of i8 pro- positions of j;^A^rrir geometry. Euclid based it on Autolycus' work y wepl tcipovfiipff^ a^loa^y but also, evidently, on an earlier textbook of Sphaerka of exclusively mathematical content. It has been con- I jectured that the latter textbook may have been due to Eudoxus^ * 8. The Optics. This book needs no description, as it has been edited by Heiberg recently*, both in its genuine form and in the recension by Theon. The Cataptrica published by Heiberg in the same volume is not genuine, and Heiberg suspects that in its present form it may be Theon 's. Il^is not even certain that Euclid wrote Cataptrica at all, as Proclus may easily have had Theon's work before him and inadvertently assigned it to Euclid^. 9. Besides the above-mentioned works, Euclid is said to have written the Elements of Music* (ai xara fiovaiK^v oTotveiworcA?). Two treatises are attributed to Euclid in our MSS. of the Musici, the KaraTOfifj mapovo^, Sectio canonis (the theory of the intervals)*, and the elaaycaiyif apfLOviKi^ (introduction to harmony). The first, resting on the Pythagorean theory of music, is mathematical and clearly and well written, the style and the form of the propositions agreeing well with what we find in the Elements, Its genuineness is confirmed not only by internal evidence but by the fact that almost the whole of the treatise (except the preface) is quoted in extenso, and Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis and attributed by him to Porphyry, but probably for the most part compiled by Pappus or some other competent mathematician'. (On the other hand Tannery set himself to prove that the treatise is not authentic'.) The second treatise is not Euclid's, but was written by Cleonides, a pupil of Aristoxenus*. Lastly, it is worth while to give the Arabians* list of Euclid's works. I take this from Suter's translation of the list of philosophers and mathematicians in the Fihrist^ the oldest authority of the kind that we possess**. "To the writings of Euclid belong further [in addition to the Elements^ : the book of Phaenomena ; the book of * Euklui'StudUn^ pp. 50 — I. ' Heiberg, op. cit, p. 46 ; Hultsch, Autolycus^ p. xii ; A. A. Bjornbo, Studien iiber i Menelaos' Stkdrik (Abhandlungen %ur GtschUhte der mathemcUischen WissenschafUn^ XI v. 1901), p. 56 sqq. 1 • Etulidis opera omnia^ vol. vii. (1895). * Heiberg, Euclid's Optics^ etc, p. 1. • Proclus, p. 69, 3. * Published in the Musici Scriptores Graeci^ ed. Jan (Teubner, 1895), pp. 113— 166. ' Jan, Musici Scriptores Gtaeci^ p. 116. * Comptes rendus de VAcad, des inscriptions et belles-lettres y Paris, 1904, pp. 439^445* Cf. Bibliotheca Mathemaiicay vi,, 1905-6, p. 325, note i. * Heiberg, Euklid'Studien, pp. 52—5; Jan, Musici Scriptores Graeci, pp. 169 — 174. *• H. Suter, Dcu McUhematiker' Verzeichniss im Fihrist in Abhandlungen %ur Geschichte dir McUkematiky VI., 189a, pp. i — 87 (see especially p. 17). Cf. Casiri, I. 339, 340, and Garti, pp. 4, 5. H. E. 2 i8 INTRODUCTION [CH, II Given Magnitudes [Data]; the book of Tones, known under the name of Music, not genuine; the book of Division, emended by Thabit; the book of Utilisations or Applications [Parisms\ not genuine; the book of the Canon ; the book of the Heavy and Light ; the book of Synthesis, not genuine; and the book of Analysis, not genuine/' It is to be observed that the Arabs already regarded the book of Tones (by which must be meant the tla^ayoryfi appi^oviKiq) as spurious. The book of Division is evidently the book on Divisions {of figures). The next book is described by Casin as " liber de utilitate suppositus." Sutcr gives reason for believing the Poristns to be meant', but does not apparently offer any explanation of why the work is supposed to be spurious. The book of the Canon is clearly the Kara-rop.ri mavov^^i. The book on *' the Heavy aVid Light '' is apparently the tract t>e itvi a pondtrosOy included in the Basel Latin translation 0^1537, and in Gregory's edition. The fragment, however, cannot safely be attributed to Euclid, for (1) we have nowhere any mention of his having written on mechanics, (2) it contains the notion of speci6c gravity in a form so clear that it could hardly be attributed to anyone earlier than Archimedes'. Suter thinks* that the works on Analysis and Synthesis (said to be spurious in the extract) may be further developments of the Data or Parisms^ or may be the interpolated proofs of EucL Xin. I — ^s» divided into analysis uj\A synt^t^is, as to which sec the notes on those propositions. ' Suter, <^« Hi, pp. 49, 50. Wenncli trAn&lai^ tb< word as "atUk." Suier i«^s that the n^rest ttiea^ning of tlie Arabic word u of "porism" is utt^ gain (NuUen, GcwionJ, while A further meaning is explan anions observation^ addition : a gain arising out of what has preceded (cT. Proclus' dentition of the porism in the sense of a corollary). * Heiberg, Euklid-Seudifn, pp. 9, jo» * Suter, ^, cU, p. 50. CHAPTER III. GREEK COMMENTATORS ON THE ELEMENTS OTHER THAN PROCLUS. That there was no lack of commentaries on the Elements before the time of Proclus is evident from the terms in which Proclus refers to them; and he leaves us in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue)^ : "Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (SiaredpvXfjTai) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean ' a figure and a platform, but not a platform and sixpfrnr***'" — In another place* he says : " Let us now turn to the elucidation of the things proved by the writer of the Elements, selecting the more subtle of the comments made on them by the ancient writers, while I cutting down their interminable diflfuseness, giving the things which are more systematic and follow scientific methods, attaching more importance to the working-out of the real subject-matter than to the variety of cases and lemmas to which we see recent writers devoting '.themselves for the most part" At the end of his commentary on Eucl. I. Proclus remarks* that ^the commentaries then in vogue were full of all sorts of confusion, and K:ontained no account of causes, no dialectical discrimination, and no (philosophic thought. These passages and two others in which Proclus refers to "the •commentators'" suggest that these commentators were numerous. He does not however give many names; and no doubt the only mportant commentaries were those of Heron, Porphyry, and Pappus. 1 Proclus, p. 84, 8. ' i.e. we retxih a certain height, use the platform so attained' as a base on which to build nother stage, then use that as a base and so on. * Proclus, p. 100, 10. * idid. p. 433, 15. ' iduf, p. 389, 11 ; p. 338, 16. 2 — 2 M INTRODUCTION [cK. ill I. Heron* Proclus alludes to Heron twice as Heron ntechaniois^, in another place* he associates him with Ctesibius. and in the three other passages* where Heron is mentioned there is no reason to doubt that the same person is meant, namely Heron of Alexandria. The date of Heron is still a vexed question, though the possible limits appear to have been practically narrowed down to the 150 years between (say) 50 B,c, and 100 A.U Martin* concluded that Heron lived till the middle of the first century B.C^ Hultsch* placed him at the end of the secoqd century B*C, Cantor in his first two editions took a middle course and gave 100 B.C as the date when he flourished*. But it is now certain that in his Mec/uinks, preserved in the Arabic and recently published ^ Heron quotes Posidonius the Stoic (of Apamea, Cicero's teacher) by name as the author of a definition of the centre of gravity. Now Posidonius lived till about the middle of the first century B.C. * and, assuming that his writings dated from not earlier than 90 or 80 B,C, we must put Heron at all events (say) fifty years later than Hultsch placed him. Cantor now, while main- taining that he belong^ to the first century B.C, admits that he may have flourished as late as the last third of it*. But in the meantime an entirely different view was elaborated by W* Schmidt, the editor of the first volume of the new edition of Heron's complete works, who assigned him to the second half of the first century a,d.* The arguments for the Urminus post gmm are mainly these* (i) Vitruvius gives in the preface to Book vii, of his De ArchiUctura (brought out apparently 14 B.C:) a list of authorities on machinatiofies from whom he had made extracts. This list contains twelve names and has every appearance of being scrupulously com- plete ; but^ while it includes Archytas, Archimedes, Ctesibius, and Fhilo of Byzantium (who come second, third, fourth, and sixth in order respectively), Heron is not mentioned. Moreover the pointii of diflfcrence between Vitruvius and Heron seem on the whole to be more numerous and important than the resemblances, (2) Diels concluded from the use of Latinisms by Heron that the first century a»D. was the earliest possible date. (3) A definite date was derived by Carra de Vaux from the identification of a small single-screw olive press described by Heron {Ah^c/mnicSj llh 20) with one mentioned by Pliny {Nat Hist XViil. 517) as having been introduced within the last twenty- two years : this gives A.D. 55 as the date before which the Mechanics could not have been written. The tenninus ante qnetn, 100 A.D., was arrived at (i) from internal evidence suggesting that ^ Proclus, p. 305, 24; p, 34^5, J3» * iMd. p, 41^ to. ' itid. p. 196, 16; p, 313, 71 p. 419, 13. * Martin, Reckfri/ui sur la xnt tt Its suvrngn d^Hfr&n d^AU^^mdrity Parish 1854, p* ^7, ' Hultsch, Meir^hgum^m icriptorum reliqumtt 1&64, 1. 9. * Cantorf Gtuh. d, Math^ i^, p, 347. ' Hsrm^is AUxandrini cpera quae tupertuni pmnia ^Teabner, Leipzig, vol* JL edited b* L* Nix and W- Schmidt, 1900. ' Cantor, Gtsch. d. Matk. I,, p. 366. * See ff^rpmi AUxattdnm p/era, vol 1-, 1899, pp, bt— xxv. ■ CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS ax Heron was earlier than Claudius Ptolemy (about 100-178 A.D.), and (2) from an apparent reference by Plutarch to a proposition about incidence and reflexion taking place at equal angles, proved by Heron in his Catoptricay coupled with the facts that in that work Heron mentions Menelaus of Alexandria (about 100 AD.) and that Plutarch died at a great age in 1 20 AD. Attempts have however been made in two recent tracts to over- throw almost the whole of these arguments ^ (i) It is asserted that the olive-press of Mechanics III. 20 is not the same as that referred to by Pliny. (2) It is pointed out that Heron is mentioned with Archimedes and Ctesibius in a passage of Proclus which is supposed to be drawn from Geminus*. But, as Geminus wrote about 70 B.C and Posidonius not earlier than 90 B.C., while Heron quotes Posidonius and is therefore later, the intervals are all too short to make it probable that Heron would be mentioned in Geminus' historical work ; and I think that the name of Heron may well have been inserted after that of Ctesibius by Proclus himself. (3) The view that Vitruvius did not use Heron's work is attacked, and the contrary sought to be proved, on the basis apparently of three passages, {a) Vitruvius* water-organ is held to be decidedly better than Heron's* : therefore Vitruvius used Heron's in order to improve upon it. {b) Vitruvius, in a passage describing a certain use of the lever, takes a wrong point to be the fulcrum ; and it is held that he cannot have made the mistake himself, but must necessarily have copied it from Heron^ In order, however, to find the same error in Heron, Hoppe arbitrarily alters both the figure and the text (c) Vitruvius describes the working of a certain crane in language less clear than that of Heron*; therefore he used Heron but misunderstood him ! AH would appear to be grist which comes to the mill of such critics : but I doubt whether such arguments will convince those who hold to the second half of the first century as the date that their view is mistaken. That Heron wrote a systematic commentary on the Elements might be inferred from Proclus, but it is rendered quite certain by references to the commentary in Arabian writers, and particularly in an-Nairizi's commentary on the first ten Books of the Elements. The Fihrist says, under Euclid, that " Heron wrote a commentary on this book [the Elements], endeavouring to solve its difficulties*"; and under Heron, " He wrote : the book of explanation of the obscurities in Euclid'...." An-Nairlzl's commentary quotes Heron by name very frequently, and often in such a way as to leave no doubt that the author had Heron's work actually before him. Thus the extracts are ^ E. Hoppe, Ein Bdtrag tur Zeithatimmung Herons von AUxandrienf Hamburg, 1909 ; Rudolf Meier, De Heronis atUUe^ Leipzig, 1905. See the references to the arguments in Cantor, Gesch. d. Math, i„ pp. 365, 367, 545—7. ' Proclus, p. 41, la ' Vitruvius, x. 13 ; Heron, vol. i. p. 19a sqq. (PntuwuUus^ I. 43, 43). ^ Vitruvius, X. 3, 3; Heron, vol. ii. pp. 114 — 116 (Mechanics^ 1 1. 8). ■ Vitruvius, X. 3, 10 ; Heron, vol. ii. pp. 101 — 4 Qifechanus^ ill. a). * DcLs MaikimoHker- Veneuhniss im Fikrist (tr. Suter), p. 16. ^ ibid, p. as. » INTRODUCTION [ch, hi given in the first person introduced by " Heron says *' (" Dixit Yrinus " or "Heron"); and in other places we are told that Heron '*says nothing/* or "is not found to have said anything " on such and such a proposition. The commentary of an-Nair!zi is being published by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Hajjaj with the commentary attached \ But this MS. only contains six Books, and several pages in the first Book are missing> which contain the comments of Simphcius on the first twenty- two definitions of the first Book, Fortunately the commentary of an-Nairizi has been discovered in a more complete form, in a Latin translation by Gherardus Cremonensis of the twelfth century, which contains the missing comments by Simplicius and an-Nairizl's com- ments on the first ten Books. This valuable work has recently been edited by Curtzel Thus from the three sources, Prod us, and the two versions of an-NairlzI, which supplement one another, we are able to form a very good idea of the character of Heron's commentary. In some cases observations given by Proclus without the name of their author are seen from an-Nairlzl to be Heron's ; in a few cases notes attributed by Proclus to Heron are found in an-Nairizi without Heron's name ; and J curiously enough, one alternative proof (of L 25) given as Heron's by Proclus is introduced by the Arab with the remark that he has not been able to discover who is the author. Speaking generally, the comments of Heron do not seem to have contained much that can be called important We find (i) A few general notes, eg. that Heron would not admit more than three axioms. (2) Distinctions of a number of particular cases of Euclid's pro- positions according as the figure is drawn in one way or in another. Of this class are the different cases of L 35, 36, III, 7, 8 (where the chords to be compared are drawn on different sides of the diameter instead of on the same side), III. 12 (which is not Euclid's, but Heron's own, adding the case of external contact to that of internal contact in III, 11), VL 19 (where the triangle in which an additional line is drawn is taken to be the smaller of the two), vii, 19 (where he gives the particular case of three numbers in continued proportion^ instead of four proportionals). (3) Alternative proofs. Of these there should be mentioned {a) the proofs of II. i — 10 " without a figure/' being simply the algebraic forms of proof, easy but uninstructive, which are so popular nowadays^ the proof of 11 1, 25 (placed after 11 L 30 and starting from the arc instead of the chord), IIL 10 (proved by llh 9), m, 13 (a proof preceded by a lemma to the effect that a straight line cannot meet a circle in more than two points). Another class of alternative proof is ^ CodAX Lddtmis 399, t, Euclidis Ekmmia ex iHt£rpr^iUi<mt ai-Hadstkdickndschii cum amrturUariii ai-Naruii. Two parts carrying the work to the end of Book I, were tssQed in 189^ and 1897 lespectively' Another part came out in 1905. * Anariitt in decern iibros £rwrts iiemeniarum Euciidu t&mmeniarii tx mUfprd^xUone Ghm^rdi Cremonensis.., aUdU Maxuniliaous Curtze (Teubner, Lelpdg^ 'S99)< I n CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS 23 {b) that which is intended to meet a particular objection (h^tmuri^) which had been or might be raised to Euclid's construction. Thus in certain cases he avoids producing a particular straight line, where Euclid produces it, in order to meet the objection of any one who should deny our right to assume that there is any space available\ Of this class are Heron's proofs of I. 1 1, 1. 20, and his note on I. 16. Similarly on I. 48 he supposes the right-angled triangle which is constructed to be constructed on the same side of the common side as the given triangle is. A third class (c) is that which avoids reductio ad absurdum. Thus, instead of indirect proofs. Heron gives direct K roofs of I. 19 (for which he requires, and gives, a preliminary jmma), and of i. 25. (4) Heron supplies certain converses of Euclid's propositions, e.g. converses of li. 12, 13, vill. 27. (5) A few additions to, and extensions of, Euclid's propositions are also found. Some are unimportant, e.g. the construction of isosceles and scalene triangles in a note on I. i, the construction of two tangents in III. 17, the remark that Vll. 3 about finding the greatest common measure of three numbers can be applied to as many numbers as we please (as Euclid tacitly assumes in VII. 31). The most important extension is that of ill. 20 to the case where the angle at the circumference is greater than a right angle, and the direct deduction from this extension of the result of III. 22. Interesting also are the notes on I. 37 (on I. 24 in Proclus), where Heron proves that two triangles with two sides of one equal to two sides of the other and with the included angles supplementary are equal, and compares the areas where the sum of the two included angles (one being supposed greater than the other) is less or greater than two right angles, and on I. 47, where there is a proof (depending on preliminary lemmas) of the fact that, in the figure of the proposition, the straight lines AL^ BKy CF meet in a point. After iv. 16 there is a proof that, in a regular polygon with an even number of sides, the bisector of one angle also bisects its opposite, and an enunciation of the corresponding proposition for a regular polygon with an odd number of sides. Van Pesch" gives reason for attributing to Heron certain other notes found in Proclus, viz. that they are designed to meet the same sort of points as Heron had in view jn other notes undoubtedly written by him. These are (a) alternative proofs of I. 5, I. 17, and l. 32, which avoid the producing of certain straight lines, {p) an alternative proof of I. 9 avoiding the construction of the equilateral triangle on the side of BC opposite to A ; (^r) partial converses of I. 35 — 38, starting from the equality of the areas and the fact of the parallelograms or triangles being in the same parallels, and proving that the bases are the same or equal, may also be Heron's. Van Pesch further supposes that it was in Heron's commentary that the proof by Menelaus of L 25 and the proof by Philo of i. 8 were given. ^ Cf. Prodns, 375, 7 c/ M X^oc rif rbww fi^ e^d^cu..., 389, 18 X^ci oSy nt 5rc oix Irrc ^ . rinrot.,., c * De ProclifoniibuSt Lugduni-Batavorum, 1900. 11 \l 24 INTRODUCTION [ch< hi The last reference to Heron made by an-Nairt^I occurs in the note on VI IL 27, so that the commentary of the former must at least have reached that point. II* Porphyry. The Porphyry here mentioned is of course the Neo-PIatonist who lived about 232-304 A.D. Whether he really wrote a systematic commentary on the £/anmfs is uncertain. The passages in Froclus which seem to make this probable are two in which he mentions him (i) as having demonstrated the necessity of the words "not on the same side" in the enunciation of L 14\ and (2) as having pointed out the necessity of understanding correctly the enunciation of L 26, since, if the particular injunctions as to the sides of the triangles to be taken as equal are not regarded, the student may easily fall into error'. , These passages, showing that Porphyry carefully analysed Euclid *s enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary. Further, the list of mathematicians in the Fihrist gives Porphyry as having written '*a book on the Elements," It is true that Wenrich takes this book to have been a work by Porphyry mentioned by Suidas and Proclus ( Tfuohg. Platen, \ TT^pi apymv libri IL* There is nothing of importance in the notes attributed to Potphyry by Proclus. (i) Three alternative proofs of L 20, which avoid producing a side of the triangle, are assigned to Heron and Porphyry without saying which belonged to which. If the first of the three was Heron's, I agree with van Pesch that it is more probable that the two others were both Porphyry's than that the second was Heron's and only the third Porphyry's. For they are similar in character, and the third uses a result obtained in the second*. (2) Porphyry gave an alternative proof of L iS to meet a childish objection which is supposed to require the part of ^C equal to AB to be cut off from CA and not from A C. Proclus gives a pr^iisely similar alternative proof of I, 6 to meet a similar supposed objection ; and it may well be that, though Proclus mentions no name, this proof was also Porphyry's, as van P^ch suggests*. Two other references to Porpbyry found in Proclus cannot have anything to do with commentaries on the Elements. In the first a work called the Xu^fiiKTa is quoted, while in the second a philo- sophical question is raised, in. Pappus. The references to Pappus in Proclus arc not numerous ; but we have other evidence that he wrote a commentary on the Elements, Thus a scholiast on the definitions of the Data uses the phrase " as * Proclas, pp. 197, t— 1981 10. * iMd. p. 353, 13, 14 and ihe pages prtcedjng. * Fihrist (tr, Suter), p. 9^ lo and p> 45 {note j). * Van Pesch, Di Pr&cH fantihu^ pp. 119, 130* Heibcrg assigned Ibero as above in hts Eukiid-Studun (p, t6o)t but «eems to OAve changed Ms view later. (See Be$thoni*Heibetg, C^tM LiidfHjis^ p- 93, note 3.) * Van Pesch, ^* fit, pp. 13^— j. _ ^ l^ CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS 25 Pappus says at the b^inning of his (commentary) on the loth (book) of Euclid*." Again in the Fikrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two parts*. Fragments of this still survive in a MS. described by Woepcke*, Paris. No. 952. 2 (supplement arabe de la Biblioth^que imp^riale), which contains a translation by Abu 'Uthman (beginning of loth century) of a Greek commentary on Book x. It is in two books, and there can now be no doubt that the author of the Greek commentary was Pappus*. Again Eutocius, in his note on Archimedes, On the Sphere and Cylinder!, 13, says that Pappus explained in his commentary on the Elements how to inscribe in a circle a polygon similar to a polygon inscribed in another circle ; and this would presumably come in his commentary on Book xil., just as the problem is solved in the second scholium on Eucl. xil. i. Thus Pappus' commentary on the Elements must have been pretty complete, an additional confirmation of this supposition being forthcoming in the reference of Marinus (a pupil and follower of Proclus) in his preface to the Data to "the com- mentaries of Pappus on the book'." The actual references to Pappus in Proclus are as follows: (i) On the Postulate (4) that all right angles are equal, Pappus is quoted as saying that the converse, viz. that all angles equal to a right angle are right, is not true*, since the angle included between the arcs of two semicircles which are equal, and have their diameters at right angles and terminating at one point, is equal to a right angle, but is not a right angle. (2) On the axioms Pappus is quoted as saying that, in addition to Euclid's axioms, others are on record as well (avpavaypaf^ecrOai) about unequals added to equals and equals added to unequals^; these, says Proclus, follow from the Euclidean axioms, while* others given by Pappus are involved by the definitions, namely those which assert that *' all parts of the plane and of the straight line coincide with one another," that " a point divides a straight line, a line a surface, and a surface a solid/' and that "the infinite is (obtained) in magnitudes both by addition and diminution'." ^ Euclid's Da/a, ed. Menge, p. 363. ' Fikrist (tr. Suter), p. 29. • Mhnoires prheniis h tacacUmie des sciences^ 1856, XI v. pp. 658 — 710. * Woepcke read the name of the aathor, in the title of the first book zsB .los (the dot representing a missing vowel). He quotes tdso from other MSS. (e.g. of the Ta'riJkh ai- aukamU and of the Fikrist) where he reads the name of the commentator %& B ,lis^ B,n.s OT B./,s. Woepcke takes this author to be Vtdens, and thinks it possible that he may be the same as the astrologer Vettius Valens. This Heiberg {Euk/ia-Stsuiien, pp. 169, i7o) proves to be impossible, because, while one of the Mss. quoted by Woepcke says that **B,M.s, le Roimi'' (late-Greek) was later than Claudius Ptolemy and the Fikrist says **B./,s, \e RoUmi" wrote a commentary on Ptolemy's Plcmispkaerium, Vettius Valens seems to have lived under Hadrian, and must therefore have been an eider contemporary of Ptolemv. But Suter shows (Fikrist^ p. 22 and p. 54, note 9)) that Bancs is only distin- guished from Badcs by the position of a certain dot, and Ba/os may also easily have arisen from an original Bados (there is no P in Arabic), so that Pappus must be the person meant. This is further confirmed by the fact that the Fikrist gives this author and Valens as the subjects of two separate paragraphs, attributing to the latter astrological works only. " Heiberg, Euklid-Studien^ p. 173; Euclid's Daia^ ed. Mexige, pp. 356, Hi. * Proclus, pp. 189, 190. ' ibid. p. 197, (^10. • ihid, p. 198, 3—15. 26 INTRODUCTION (3) Pappus gave a pretty proof of L 5. This proof has, I think; been wrongly understood ; on this point sec my note on the proposition. (4) On h 47 Proclus says^ : ** As the proof of the writer of the Elements is manifest, I think that it is not necessary to add anything further, but that what has been said is sufficient, since indeed those who have added more^ like Heron and Pappus, were obliged to make use of what is proved in the sixth book, without attaining any important result.'* We shall see what Heron's addition consisted of; what Pappus may have added we do not know, unless it was some- thing on the lines of his extension of L 47 found in the Syfu^ge (iv/p. 176, ed. Hultsch). We may fairly conclude, with van Pesch*, that Pappus is drawn upon in various other passages of Proclus where he quotes no authority, but where the subject-matter reminds us of other notes expressly assigned to Pappus or of what we otherwise know to have been favourite questions with him. Thus : r We are reminded of the curvilineal angle which is equal to but not a right angle by the note on I. 32 to the efllcct that the converse (that a figure with its interior angles together equal to two right angles is a triangle) is not true unless we confine ourselves to rectilineal figures. This statement is supported by reference to a figure formed by four semicircles whose diameters form a square, and one of which is turned inwards while the others are turned outwards. The figure forms two angles "equal to" right angles in the sense described by Pappus on Post 4, while the other curvilineal angles are not considered to be angles at all, and are left out in summing the internal angles. Similarly the allusions in the notes on L 4, 23 to curvilineal angles of which certain moon-shaped angles {p.i}PQ€i&€h) are shown to be "equal to*' rectilineal angles savour of Pappus, 2, On L 9 Proclus says* that ** Others, starting from the Archi- medean spirals, divided any given rectilineal angle in any given ratio." We cannot but compare this with Pappus IV. p, 286, where the spiral is so used ; hence this note, including remarks immediately preceding about the conchoid and the quadratrix, which were used for the same purpose, may very well be due to Pappus. 3, The subject of isopen metric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subject* Now on L 35 Proclus speaks* about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelo- grams with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on I. 37 about triangles* Compare' also his note on l. 4. These notes may have been taken from Pappus. * Proclus p, 419, g— '5- ' Van Pcsch, Dt Procli f^rUibtt^^ y, 134 sqq. * Proclus, p. 171, jo* * Pappu^ V. pp. 304—^50; for Zenodorus own treatise see HuJ tech's Appendix, pp. it 89 — i*n* * Produ% pp. 396 — 8. • iMd. pp. 403 — 4- '' iHd. pp. 336 — 7, t CH-Jii] GREEK COMMENTATORS OTHER THAN PROCLUS a? 4. Again, on I. 21, Proclus remarks on the paradox that straight lines may be drawn from the base to a point within a triangle which are (i) together greater than the two sides, and (2) include a less angle provided that the straight lines may be drawn from points in the base other than its extremities. The subject of straight lines satisfying condition (i) was treated at length, with reference to a variety of cases, by Pappus*, after a collection of "paradoxes" by Erycinus, of whom nothing more is known. Proclus gives Pappus' first case, and adds a rather useless proof of the possibility of drawing straight lines satisfying condition (2) alone, adding that " the proposi- tion stated has been proved by me without using the parallels of the commentators'." By " the commentators " Pappus is doubtless meant 5. Lastly, the "four-sided triangle," called by Zenodorus the ••hollow-angled,"* is mentioned in the notes on I. Def. 24 — 29 and I. 21. As Pappus wrote on Zenodorus' work in which the term occurred^ Pappus may be responsible for these notes. IV. Simplicius. According to the Fihrist^, Simplicius the Greek wrote ''a com- mentary to the beginning of Euclid's book, which forms an introduc- tion to geometry." And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the Parallel- Axiom) is preserved in the Arabic commentary of an-Nairizi*. On two subjects this commentary of Simplicius quotes a certain " Aganis," the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-NairlzI after I. 29, an attempt by •• Aganis " to prove the parallel-postulate. It starts from a definition of parallels which agrees with Geminus' view of them as given by Proclus^ and is closely connected with the definition given by Posidonius*. Hence it has been assumed that "Aganis" is none other than Geminus, and the historical importance of the commentary of Simplicius has been judged accordingly. But it has been recently shown by Tannery that the identification of " Aganis " with Geminus is practically impossible*. In the translation of Besthom-Heiberg Aganis is called by Simplicius in one place " philosophus Aganis," in another " magister noster Aganis," in Gherard's version he is " socius Aganis" and "socius noster Aganis." These expressions seem to leave no doubt that Aganis was a contemporary and friend, if not master, of Simplicius ; and it is impossible to suppose that Simplicius (fl. about 500 A.D.) could have used them of a man who lived four and • Pappus III. jM). 104—130. » Proclus, p. 318, 15. ' Proclus, p. 105, 14; cf. pp. 318, 339. * See Pappus, ed. Hultsch, pp. 11 54, 1306. • Fihrist (tr. Suter), p. ii. • An-NairizI, cd. Bestbom-Heiberg, pp. 9—41, 119;— 1331 cd. Curtze, pp. i — 37, 65 — 73. The Codix LHdensiSy from which ^sthom and Heiberg are editing the work, has un- fortunately lost some leaves so that there is a gap from Def. i to Def. 35 (parallels). The loss is, however, made good by Curtze's edition of the translation by Ghenmi of Cremona. ' Proclus, p. 177, 31. • ibid, p. 176, 7. ' Biblioikeca Mathematical ii^, 1900, pp. 9 — 11. "1 tS INTRODUCTION [ciL m J a half centuries before his time. A phrase in Simplicius* word-for- word quotation from Aganis leads to the same conclusion. He speaks of people who objected *'cven in ancient times" (iam antiquitus) to ^ the use by geometers of this postulate. This would not have been an | appropriate phrase had Gcminus been the writer I do not think that this difficulty can be got over by Suter's suggestion* that the passages in question may have been taken out of Heron's commentary, and that an-Nairizi may have forgotten to name the author ; it seems clear that Simplicius is the person who described " Aganis." Hence we are driven to suppose that Aganis was not Geminus, but some unknown contemporary of Simplicius*, Considerable interest will however continue to attach to the comments of Simplicius so fortunately preserved* Proclus tells us that one Aegaeas {? Aenaeas) of Hierapolis wrote an epitome of the Ekments^ ; but we know nothing more of him or of it 1 Z€its€krififur Ai^th. u. Pkysik, XLIV., hLit.-tttt Abtb Jf. 6j* ^ The ibove Aripimeni seems Lome quUc infiuperahte. Ine other Ai^inents of Tvunery do Dot* however, carry conviction to my mind* I dci ttot follow the reaM>niiig bailed on Aganis' definition of an angle. It appears to me a pure assumption ihat Geminus would have seen that Po^idonius^ definition of pftralleb w^ not admissible. Nor 6ot^ it seem to me to count for mych that Proclus, while telling us that Geminus held that the postulate ought to be proved and warned the unwaiy again^ hastUy concluding that two straight lines approaching one another mu&t necci^sarity meet (cf« a curve and it& asymptote), gives no nint that Geminus did try to prove the postulate^ It may well be tnat Procltis omitted Getninns* "proofs (if he trrote one) because he preferred Ptolemy*s attempt which he give (PP- 3^5—7)' * Proclus, p. 361* 11* CHAPTER IV. PROCLUS AND HIS SOURCES^. It is well known that the commentary of Proclus on Eucl. Book I. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio. Proclus himself lived 410-485 A.D., so that there had already passed a sufficient amount of time for the tradition relating to the pre-Euclidean geometers to become obscure and defective. In this connexion a passage is quoted from Simplicius' who, in his account of the quadrature of certain lunes by Hippocrates of Chios, while mentioning two authorities for his statements, Alexander Aphro- disiensis (about 220 A.D.) and Eudemus, says in one place', ''As regards Hippocrates of Chios we must pay more attention to Eudemus, sittce he was nearer the times, being a pupil of Aristotle." The importance therefore of a critical examination of Proclus' commentary with a view to determining from what original sources he drew need not be further emphasised. Proclus received his early training in Alexandria, where Olympio- dorus was his instructor in the works of Aristotle, and mathematics was taught him by one Heron* (of course a different Heron from the '^mechanicus Hero" of whom we have already spoken). He after- wards went to Athens where he was imbued by Plutarch, and by Syrianus, with the Neo-Platonic philosophy, to which he then devoted ^ My task in this chapter is made easy by the appearance, in the nick of time, of the dissertation De Prodi fotUitms by J. G. van Pesch (Lugduni-Batavorum, Apud L. van Nifterik, MDCCCC). The chapters dealing directly with the subject show a thorough acquaintance on the part of the author with all the literature bearing on it; he covers the whole field and he exercises a sound and sober judgment in forming his conclusions. The same cannot always be said of his only predecessor in the same inquiry, Tannery (in La GhmHrie grtcque, 1887), ^^^ often robs his speculations of much of their value through his proneness to run away with an idea ; he does so in this case, basing most of his conclusions on an arbitrary and unwarranted assumption as to the significance of the words ol repf rua (e.g.'Upfura, no^ctdt&rtov etc) as used in Proclus. • Simplicius on Aristotle*s Physics^ ed. Dieb, pp. 54 — 69. * ibid, p. 6%, 31. ^ Cf. Martin, Recktrcha sur lavidet Us omfrages d^Hiron dAUxandrii, pp. 340 — 3. 30 INTRODUCTION [ch. iv heart and soul, becoming one of its most prominent exponents* He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contem- poraries, as we learn from Marinus his pupil and biographer. On the death of Syrian us he was put at the head of the Neo-Platonic school He was a man of untiring industry^ as is shown by the number of books which he WTote, including a large number of com- mentaries, mostly on the dialogues of Plato* He was an acute dialectician, and pre- eminent among his contemporaries in the range of his learning^; he was a competent mathematician; he was even a poet At the same time he was a believer in all sorts of myths and mysteries and a devout worshipper of divinities both Greek and Oriental, Though he was a competent mathematician, he was evidently 4nuch more a philosopher than a mathematician I This is shown even in his commentary on EucL !.» where, not only in the Prologues (especially the first^ but also in the notes themselves, he seizes any opportunity for a philosophical digression. He says himself that he attaches most importance to **the things which require deeper study and contribute to the sum of philosophy*"; alternative proofs, cases ^ and the like (though he gives many) have no attraction for him ; and, in particular, he attaches no value to the addition of Heron to L 47*, which is of considerable mathematical interest Though he esteemed mathematics highly, it was only as a handmaid to philosophy. He quotes Plato's opinion to the effect that "mathematics, as making use of hypotheses, falls short of the non-hypothetical and perfect science*". »/' Let us then not say that Plato excludes mathematics from the sciences, but that he declares it to be secondary to the one supreme science V And again, while "mathematical science must be considered desirable in itself, though not with reference to the needs of daily lifej'' **if it is necessary to refer the benefit arising from it to , something else, we must connect that benefit with intellectual know- ledge {voepcLv yptii>a-tp)y to which it leads the way and is a propaedeutic, clearing the eye of the soul and takii)g away the impediments which the senses place in the way of the knowledge of universals {twv Skmvyr We know that in the Neo* PI atonic school the younger pupils learnt mathematics ; and it is clear that Proclus taught this subject, and that this was the origin of the commentary. Many passages show him as a master speaking to scholars. Thus "we have illustrated ^ Edict c^ Is him '* Der GelchrtCi dem kein Feld damaligen Wissens verschlos^cn ist/^ ' Van Tesch observes that in his commentaries on the Tiwa^iw (pp* 67;— 1> he fipe^ks as no real maihemaiician could h^ve spoken. In the passage referred Lo ihe qu^uon h whether the &un occupies a middie place among the planets^ Proclus rejects the view of Hipparchus and Ptolemy because **d &toupy6v^^ ($c. the Chaldean, says Zellcr) thinks otherwise , "whom it is not lawful to disbeJieve/' Martin says rather neatly^ *^ Pour ProduSi les Elements d'EucUde ont Thcnrcu^e chance de n'^tre contr«dits ni par fei Omclcs cbaldaSfques, tu par tes speculations des pythtigoneicns ancicns et nouveaux......" • Proclus, p* S41 13* ' iHd. p, 419^ Up * ibid^ p. 3t, 30» • ibid, p. %t, %" i ^ itid. p. a;, a; to aS, 7; cf. alio p. 11, 95, pp» 46, 47. CH. TV] PROCLUS AND HIS SOURCES 31 and made plain all these things in the case of the first problem, but it is necessary that my hearers should make the same inquiry as regards the others as welP," and '' I do not indicate these things as a merely incidental matter but as preparing us beforehand for the doctrine of the Timaeusl" Further, the pupils whom he was I • addressing were beginners in mathematics ; for in one place he says ^ that he omits " for the present" to speak of the discoveries of those I who employed the curves of Nicomedes and Hippias for trisecting I an angle, and of those who used the Archimedean spiral for dividing an angle in any given ratio, because these things would be too difficult for beginners (SvaOeapiJTovi roU elaayofiepoi^y. Again, if his pupils had not been beginners, it would not have been necessary for Proclus to explain what is meant by saying that sides subtend certain angles^ the difference between adjacent and vertical angles' etc., or to exhort them, as he often does, to work out other particular cases for themselves, for practice {yviiva^ria^ ev€Ka)\ The commentary seems then to have been founded on Proclus' lectures to beginners in mathematics. But there are signs that it was revised and re-edited for a larger public ; thus he gives notice in one place' "to those who shall come upon" his work (to*9 ivrev^o- fihoi^y There are also passages which could not have been under- stood by the beginners to whom he lectured, e.g. passages about the cylindrical helix', conchoids and cissoids'. These passages may have been added in the revised edition, or, as van Pesch conjectures, the explanations given in the lectures may have been much fuller and more comprehensible to beginners, and they may have been shortened on revision. In his comments on the propositions of Euclid, Proclus generally proceeds in this way : first he gives explanations r^arding Euclid's proofs, secondly he gives a few different cases, mainly for the sake of practice, and thirdly he addresses himself to refuting objections raised by cavillers to particular propositions. The latter class of note he deems necessary because of "sophistical cavils" and the attitude of the people who rejoiced in finding paralogisms and in causing annoyance to scientific men". His commentary does not seem to have been written for the purpose of correcting or improving Euclid. For there are very few passages of mathematical content in which Proclus can be supposed to be propounding anythirtg of his own ; nearly all are taken from the works of others, mostly earlier commentators, so that, for the purpose of improving on or correcting Euclid, there was no need for his commentary at all. Indeed only in one place does he definitely bring forward anything of his own to get over a difficulty which he finds in Euclid"; this is where he tries to • Proclus, p. a 10, 18. • iMd, p. 384, 1. • Mi, p. 171, II. * Md. p. 138, 13. • Md. p. 198, 14. • Cf. p. 314, 15 (on 1. 1). ' iHd. p. 84, g. • idtd, p. 105. 9 .•A.W ^ .7^ 10 .A.V/ n -.fe ft wvut, ^* 04, y* ' Md. p. 113. " Md. pp. 368—373. *• lAfV/. p. 375, 8. i- 3a INTRODUCTION [ch. iv prove the parallel-postulate, after first giving Ptolemy's attempt and then pointing out objections to it. On the other hand there are a number of passages in which he extols Euclid ; thrice* also he supports Euclid against Apollonius where the laUer had given proofs which he considered better than Euclid's (I. lo, ii, and 23). Allusion must be made to the debated question whether Proclus continued his commentaries beyond Book L His intention to do so is clear from the following passages. Just after the words above quQted about the trisection etc* of an angle by means of certain curves he says, '* For we may perhaps more appropriately examine these things on the third book, where the writer of the Elements bisects a given circumference^" Again, after saying that of all paraJlelograms which have the same perimeter the square is the greatest *' and the rhomboid least of all/' he adds: "But this we wilt prove in another place; for it is more appropriate to the (discussion of the) hypotheses of the second book* " Lastly, when alluding (on h 45) to the squaring of the circle, and to Archimedes' proposition that any circle is equal to the right-angled triangle in which the perpendicular is equal to the radius of the circle and the base to its perimeter, he adds, '' But of this elsewhere**'; this may imply an intention to treat of the subject on Eucl. XIL, though Heiberg doubts it** But it is clear that, at the time when the commentary on Book I. was written* Proclus had not yet b^^n to write on the other Books and was uncertain whether he would be able to do so : for at the end he says\ '* For my part, if I should be able to discuss the other books^ in the same manner, I should give thanks to the gods ; but, if other cares should draw me away, I beg those who are attracted by this subject to complete the exposition of the other books as well, following the same method, and addressing themselves throughout to the deeper and better denned questions involved '' (to 'Trpayfjt^aretwS^^ vapra^ov xal ev&iaiperov fLeraStoi Kovras:)- There is in fact no satisfactory evidence that Proclus did actually write any more commentaries than that on Book J*, those who have attributed to Proclus some of the scholia on the later books having failed to prove their case\ The contrary view receives support from two facts pointed out by Heiberg, viz. (i) that the scholiast's copy of Proclus was not much better than our MSS. : in particular, it had * Proclus, p. aSo, gj p. 183* 10; pp, 335, $36, * i6id. p» 371, 14* ■ i^J. p. J98, 1 8, * ihui, p, 433, 6. • Heiberg J £uklid*Studi£n, p* 165, note. ' Proclus, p. +33* 9. ^ The words in ihe Greek arc: ti ptHv Sifwi^Btlnifiet x^i roTt XiHiro7» m ovr^ rpi/wof ^\$w. For i^t\9ttif Heiberg would read iwf^fMtty. ' Heiberg {EuMUd-Sindkn, p- 166) gives reason for doubling the evidoice mdduoed by Waclumutht by which Knoche vras persuaded to give up his ori^nal view (ha.t Proclus did nai write any more comtnentaries. Wacbimutb tie lies sole-ly upon a Vatican MS- which ha>i at the head of a collection of scholia on Books K (extracts from the e sclent convmcnUry of Proclus), IL, Vm VI., 9C. the title ; Eti ri EujfXff^n; ^rwx^ia wpo\aft^iu/6fiiPa in tQv Iip6ii\o*t €wopdhfw jfol jcat' iwiT^^. Hcibeig holds Umt this liile its^Jf makes ti probable that the anthor^ip ascribed to Proclus was restricted to the scholia on Book J.: otherwise how could one understand the expression irpoXati^ti^btitwa in tmp llp6K\ov^ which words would suit extracts ^m Frndtu'/fW^s^tfi weU enough, but not the acholia to latei^ Books? CH. nr] PROCLUS AND HIS SOURCES 33 the same lacunae in the notes on l. 36, 37 and 1. 41 — ^43 : and even this fact makes it improbable that the scholiast had further commentaries of Proclus which have vanished for us ; (2) that there is no trace in the scholia of the notes which Proclus promised in the passages quoted above \ Coming now to the question of the sources of Proclus, we may say that everything goes to show that his commentary is a compilation, though a compilation "in the better sense" of the term*. He does not even give us to understand that we shall find in it much of his own ; " let us," he says, " now turn to the exposition of the theorems proved by Euclid, selecting the more subtle of the comments made on them by the ancient writers, and cutting down their interminable difluse- ness...*'': not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary ; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times ; but we now know, from the commentary of an-NairlzI, that a number of other passages, where he mentions no name, are taken from Heron, and among them the not unimportant addition of an alternative proof to I. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is generally the other way ; and it is often possible to arrive at a con- clusion, either that a particular note is not Proclus' own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some com- mentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentator^ Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible ; the note is that in which a converse of I. 32 is proved*, the proposition namely that a rectilineal figure which has all its interior angles together equal to two right angles is a triangle. It is not safe to attribute a passage to Proclus himself because he uses the first person in such expressions as " I say " or " I will prove '* — for he was in the habit of putting into his own words the substance of notes borrowed from others — nor because, in speaking of an 1 Heiberg, Euklid'Studim, pp. 167, 168. ' Kooche, UfUersuchungen iiber des Froklus Diadochus Comnuntar tu Euklitts EU- menien (1863) p. 11. ■ Proclus, p. aoo, 10 — 15. * Instances of the application of this criterion will be found in the discussion of Proclus* indebtedness to the commentaries of Heron, Porphyry and Pappus. * Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the proof is followed by a passage which, on comparison with Pappu? note on the postulate that all right angles are equal, he feeb justified in assigning to Pappus. I doubt if ttie evidence is sufficient. H. K. 3 U INTRODUCTION [cH. nr objection raised to a particular proposition, he uses such expressions as "perhaps someone may object" (fir<w? S* av rtj^cT ipirraUp^^*): for sometimes other words in the same passage indicate that the objection had actually been taken by someone'. Speaking generally, we shall not be justified in concluding that Proclus is stating something new of his own unless he indicates this himself in express terms. As regards the form of Proclus* references to others by name, van Pesch notes that he very seldom mentions the particular wark from which he is borrowing. If we leave out of account the references to Plato's dialogues, there are only the following references to books : tiieBac^ia^ of Philolaus\ the Symmikta of Porphyry", Archimedes Oh the Sphere and Cyiindtr^^ Apollonius On the cochiias^^ a book by Eudemus on The Ang'ie\ a whole book of Posidonius directed against Zeno of the Epicurean sect^ Carpus* Astrofwmy^, Eudemus' History of Geometry^ ^ and a tract by Ptolemy on the parallel -postulate". Again, Proclus does not always indicate that he is quoting some- thing at second-hand. He often does so, e.g. he quotes Heron as the authority for a statement about PhiJippus, Eudemus as attributing a certain theorem to Oenopides etc.; but he says on h \2 that "Oeno- pides first investigated this problem, thinking it useful for astronomy** . when he cannot have had Oenopides* work before him. i It has been said above that Proclus was in the habit of stating in * his own words the substance of the things which he borrowed. We j are prepared for this when we find him stating that he will select the best things from ancient commentaries and "cut short their intermin- able diffuseness/* that he will * briefiy describe '* {avvrofi^K; itrTopTJaai) the other proofs of I. 20 given by Heron and Porphyry and also the proofs of I, 25 by M^nelaus and Heron, But the best evidence is of course to be found in the passages where he quotes works still extant, e*g. those of Plato, Aristotle and Plotinus, Examination of these passages shows great divergences from the original ; even where he purports to quote textual ly, using the expressions *' Plato saySj'' or •' Plotinus says," he by no means quotes word for word". In fact, he seems to have had a positive distaste for quoting textually from other works. He cannot conquer this even when quoting from Euclid ; he says in his note on I. 22, "we will follow the words of the geometer*' but fails, nevertheless, to reproduce the text of Euclid unchanged". We now come to the sources themselves from which Proclus drew ^ Van Pesch illustrates this by an objection refuted in the note on i. 9, p. 173, ir sqq. After using the above expression to introduce the objection, Proclus uses further on (p. 373, 15) the term **they say" {^aaU^). » Proclus, p. 11, 15. ' idiit. p. 56, 15. * Mt.p. 71, 18. • tiiJ, p. J05, 5. * idut. p. 135, 8. ' s'h'd, p. 300, 3. * sHd, p. 341, 19. • i^ut. p. 353, 15. " Md. p. 363, 15. " See the passages referred to by van Pesch (p. 70). The most glaring case is a passage (p. 31, 19) where he quotes Plotinus, using the expression *' Plotinus says " Comparison with Plotinus, Ennead, I. 3. 3i shows that very few words are those of Plotinus himself; the rest represent Plotinus* views in Proclus' own language. " Proclus, p. 330, 19 sqq. r CH. iv] PROCLUS AND HIS SOURCES 35 in writing his commentary. Three have already been disposed of, viz. Heron, Porphyry and Pappus, who had all written commentaries on the Elements^, We go on to Eudemus, the pupil of Aristotle, who, among other works, wrote a history of arithmetic, a history of astronomy, and a history of geometry. The importance of the last mentioned work is attested by the frequent use made of it by ancient writers. That there was no other history of geometry written after the time of Eudemus seems to be proved by the remark of Proclus in the course of his famous summary : " Those who compiled histories bring the development of this science up to this point Not much younger than tliese is Euclid*...." The loss of Eudemus' history is one of the gravest which fate has inflicted upon us, for it cannot be doubted that Eudemus had before him a number of the actual works of earlier geometers, which, as before observed, seem to have vanished completely when they were superseded by the treatises of Euclid, Archimedes and Apollonius. As it is, we have to be thankful for the fragments from Eudemus which such writers as Proclus have preserved to us. 1 agree with van Pesch* that there is no sufficient reason for doubting that the work of Eudemus was accessible to Proclus at first hand. For the later writers Simplicius and Eutocius refer to it in terms such as leave no room for doubt that thty had it before them. I have already quoted a passage from Simplicius' account of the lunes of Hippocrates to the effect that Eudemus must be considered the best authority since he lived nearer the times^ In the same place Simplicius says', ** I will set out what Eudemus says word for word (xariL Xi^ip Xeyofi^va), adding only a little explanation in the shape of reference to Euclid's Elements owing to the memorandum-like style of Eudemus {h^a top vwofipnfiaTMOP rpoirop rov EvSif/xou) who sets out his explanations in the abbreviated form usual with ancient writers. Now in the second book of the history of geometry he writes as follows'." It is not possible to suppose that Simplicius would have written in this way about the style of Eudemus if he had merely been copying certain passages second-hand out of some other author and had not the original work itself to refer to. In like manner, Eutocius speaks of the paralogisms handed down in connexion with the attempts of Hippocrates and Antiphon to square the circle', "with which I imagine that those are accurately acquainted who have examined {iirecKefifiipov^) the geometrical history of Eudemus and know the Ceria Aristotelica." How could the contemporaries of Euto- cius have examified the work of Eudemus unless it was still extant in his time ? The passages in which Proclus quotes Eudemus by name as his authority are as follows : (i) On I. 26 he says that Eudemus in his history of geometry ^ See pp. 10 to 17 above. • Proclus, p. 68, 4 — 7. ' De Proclifontibus, pp. 73 — 75. ^ See above» p. 19. ^ Simplicius, loc. cit., ed. Diels, p. 60, 17. • Archimedes, ed. Heiberg, vol. in. p. 164. 36 INTRODUCTION [ch, iv referred this theorem to Thales, inasmuch as it was necessary to Thales method of ascertaining the distance of ships from the shore', (2) Eudemus attributed to Thales the discovery of Eucl L 15', and (3) to Oenopides the problem of 1. 23". (4) Eudemus referred the discovery of the theorem in L 32 to the Pythagoreans, and gave their proof of it^ which Proclus reproduces*. (j) On I, 44 Proclus tells us* that Eudemus says that *' these things are ancient, being discoveries of the Pythagorean muse, the application (wapa^oXi^) of areas, their exceeding {yirfpffoXf)) and their falling short (eXXci^K)*" The next words about the appro- priation of these terms (parabola, hyperbola and ellipse) by later writers (i.e, Apollonius) to denote the conic sections are of course not due to Eudemus. Coming now to notes where Eudemus is not named by Proclus, we may fairly conjecture, with van Pesch, that Eudemus was really the authority for the statements (i) that Thales first proved that a circle is bisected by its diameter" (though the proof by nduciio ad absurdum which follows in Proclus cannot be attributed to Thales^), (2) that '* Plato made over to Leodamas the analytical method, by means of which it is recorded (/txTopi^ra*) that the latter too made many discoveries in geometry V* (3) that the theorem of I* 5 was due to Thales, and that for equal angles he used the more archaic expression "similar'' angles*, (4) that Oenopides first investigated the problem of L 12. and that he called the perpendicular the gfiomonic line {Kara yva^fioi'ay^, (5) that the theorem that only three sorts of polygons can fill up the space round a point, viz. the equilateral triangle, the square and the regular hexagon, was Pythagorean", Eudemus may also be the authority for Proclus' description of the two methods, referred to Plato and Pythagoras respectively, of forming right-angled triangles in w^hole numbers". We cannot attribute to Eudemus the beginning of the note on I, 47 where Proclus says that ''if we listen to those who like to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery"/' As such a sacrifice was contrary to the Pytha- gorean tenets, and Eudemus could not have been unaware of this, the story cannot rest on his authority. Moreover Proclus speaks as though he were not certain of the correctness of the tradition ; indeed, * FrocliiS, p. 351, 14^ — 18. * i&iJ^ p. 399, 3. » iSit/. p* 333, 5. * iAid. p. 379, i— 16. * iM, p, 419, 15—18. * i^id, p. ij7, 10, It. ^ Cvitor (^^fi. d. Alaih, ij, p. %%%) poinis Qui Uic connexion between the nducii^ ad ahiurdttm and the analyticiit method said to have been discovered by Piato. Proclus gives the proof by reductio ^ abiurdum to meet an imagiitary critic who d^tres a maihtmatkal proof ; possibly Thales may have been satisRei! with the argument in the same aeotence which mentions Thales, ^Mhe cause of the bisection being the unswerving course of the straight line through the centre." * Proclus, p. 31I9 19—33. ' i^' p. 350, 30. *• Md, p. S83, 7 — 10. " iHd. pp. 304, II — 305, 3. ^ iHd. pp. 438, 7— 439, 9. ^ Hid. p. 4s6, 6—9. 1 \ r ! CH. IV] PROCLUS AND HIS SOURCES 37 so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semi- circle is a right angle*, and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areas*. Plutarch's doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus "Logisticus" (the "calculator") cited by Diogenes Laertius also*; and Proclus may have had in mind this couplet with the passages of Plutarch. We come now to the question of the famous historical summary given by Proclus*. No one appears to maintain that Eudemus is the author of even the early part of this summary in the form in which Proclus gives it It is, as is well known, divided into two distinct parts, between which comes the remark, "Those who compiled histories' bring the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, collecting many of the theorems of Eudoxus, perfecting many others by Theaetetus, and bringing to irrefragable demonstration the things which had only been somewhat loosely proved by his pre- decessors." Since Euclid was later than Eudemus, it is impossible that Eudemus can have written this. Yet the style of the summary after this point does not show any such change from that of the former portion as to suggest different authorship. The author of the earlier portion recurs frequently to the question of the origin of the elements of geometry in a way in which no one would be likely to do who was not later than Euclid ; and it must be the same hand which in the second portion connects Euclid's Elements with the work of Eudoxus and Theaetetus'. If then the summary is the work of one author, and that author not Eudemus, who is it likely to have been ? Tannery answers that it is Geminus' ; but I think, with van Pesch, that he has failed to show why it should be Geminus rather than another. And certainly the extracts which we have from Geminus' work suggest that the sort of topics which it dealt with was quite different ; they seem rather to have been general questions of the content of mathematics, and even Tannery admits that historical details could only have come inci- dentally into the work'. Could the author have been Proclus himself.^ Circumstances 1 Diogenes Laertius, i. 94, p. 6, ed. Cobet. ' Plutarch, non posse suaviter vivi secundum Epicurum, 1 1 ; Symp, vili, i. ' Diog. LAert vill. 11, p. 107, ed. Cobet: 'HrUa Iiv$ay6f^ rb ve/xjcXe^r c0pero ypdfAfia^ K^uf 4^ Ih-ti) JcXeirV ^*7* fiovOvairfp. See on this subject Tannery, La CiomJtrie grecque^ p. 105. • Proclus, pp. 64 — 70. • The plural is well explained by Tannery, La GiomHrie grecque, pp. 73, 74. No doubt the author of the summary tried to supplement Eudemus by means of any other histories which threw light on the subject. Thus e.g. the allusion (p. 64, 11) to the Nile recalls Herodotus. Cf. the expression in Proclus, p. 64, 19, vapii tup roXXQv irr^ptfrak, • Tannery, La Gioniitrie grecque^ p. 75. ' ibid, pp. 66—75. • »W. p. 19. » 38 INTRODUCTION which seem to suggest this possibility are (i^ that, as already stated, | the question of the origin of the Ekmiuts is kept prominent, , (2) that there is no mention of Democritus, whom Eudemus would not be likely to have ignored, while a follower of Plato would be likely enough to do him the injustice, following the example of Plato who was an opponent of Democntus, never once mentions him, and is said to have wished to burn all his writings ^ and (3) the allusion at the beginning to the " inspired Aristotle ** (o tat^jiovto^ 'ApiaTOT€Xf}<;)\ though this may easily have been Inserted by Proclus tn a quotation made by him from someone else On the other hand there arc considerations which suggest that Proclus himself was mt the writer, (i) The style of the whole passage is not such as to point to him { as the author. (2) If he wrote it, it is hardly conceivable that he would have passed over in silence the discovery of the analytical method, the invention of Plato to which he attached so much importance*. There is nothing improbable in the conjecture that Proclus quoted the summary from a compendium of Eudemus' history made by some later writer: but as yet the question has not been definitely settled. All that is certain is that the early part of the summary must have been made up from scattered notices found in the great work of Eudemus. > [ Proclus refers to another work of Eudemus besides the history, viz. a book on T/i£ Angle {fiiffktoi/ Trept y^vias)\ Tannery assumes that this must have been part of the history, and uses this assumption to confirm his idea that the history was arranged according to subjects, not according to chronological order*. The phraseology of Proclus however unmistakably suggests a separate work; and that the history was chrmmlogicaily arranged seems to be clearly indicated by the remark of Simplicius that Eudemus **also counted Hippocrates among the more ancient writers " (cf toX^ TraXatoripo^^)*, The passage of Simplicius about the lunes of Hippocrates throws considerable light on the style of Eudemus' history. Eudemus wrote in a memorandum -like or summary manner (top v^o^vrifiaTLKov rpGirov rov l&ifB^fiovy when reproducing what he found in the ancient writers ; sometimes it is clear that he left out altogether proofs or constructions j of things by no means easy". Geminus. The discussions about the date and birthplace of Geminus form a whole literature, for an account of which I must refer the reader to the (];• recent edition by Manitius of Gemini elementa astronamiae (Teubner, 1898)'. It must suffice here to state the general conclusion arrived at by Manitius^^ Though the name looks'^like a Latin name (Geminus), ^ Diog. Laertins, IX. 40, p. 337, ed. Cobet * Proclus, p. 64, 8. ' Proclus, p. an, 19 sqq. ; the passage is quoted above, p. 36. ^ ibid. p. 135, 8. ' Tannery, La Ghmitrie grtcque^ p. 36. * Simplicius, ed. Diels, p. 69, 33. ' ibid, p. 60, 39. * Cf. Simplicius, p. 63, 19 sqq. ; p. 64, 35 saq. ; also Usener's note " de supplendis Hippocratis quas omisit Eudemus constructionibus *' added to Diels' pre£u:e, pp. xziu — xxvL * See the appendix to this edition, pp. 337 — 353. ^* pp. 351, 353. CH. ivj PROCLUS AND HIS SOURCES 39 the consistent appearance of it in Greek with the properispomenon accent (Fe/i&ov) leaves no room for doubt as to its Greek origin, especially as it is found in inscriptions with the spelling Fc^ielvo^. The name may be formed from the stem ycfi like ^Epylvo^ from ipy, *AX€f*i/o9 from oXef. Cf also the unmistakably Greek names 'IktIpo^, Koarufo^, Geminus, a Stoic philosopher, born probably in the island of Rhodes, was the author of a comprehensive work on the classifi- cation of mathematics, and also wrote, about 73-67 B.C., a not less comprehensive commentary on the meteorological textbook of his teacher Posidonius of Rhodes. It is the former work in which we are specially interested here. Though Proclus made great use of it, he does not mention its title, unless we may suppose that, in the passage (p. 177, 24) where, after quoting from Geminus a classification of lines which never meet, he says, •* these remarks I have selected from the il>iXo/ea\la of Geminus," ^iXoxaXia is a title or an alternative title. Pappus however quotes a work of Geminus *'on the classification of the mathematics" (iv r^ w€pi T^9 r£v fiaOrffjidrmv Taf6a)9)^ while Eutocius quotes from "the sixth book of the doctrine of the mathematics " {iv r^ iKnp t§9 r&v IJLaOfifiartov OempUn;)*. Tannery* pointed out that the former title corresponds well enough to the long extract* which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae colUctiones published by Hultsch at the end of his edition of Heron* ; but it does not suit most of the other passages borrowed by Proclus. The correct title was therefore probably that given by Eutocius, The Doctrine, or Theory, of the Mathematics ; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conies, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later. If again Geminus finished his work and wrote with the same fulness about the other branches of mathematics as he did about geometry, there must ha(re been a considerable number of Books altogether. At all events it seems to have been designed to give a complete view of the whole science of mathematics, and in fact to be a sort of encyclopaedia of the subject. I shall now indicate first the certain, and secondly the probable, obligations of Proclus to Geminus, in which task I have only to follow van Pesch, who has embodied the results of Tittel's similar inquiry also*. I shall only omit the passages as regards which a case for attributing them to Geminus does not seem to me to have been made out. First come the following passages which must be attributed to Geminus, because Proclus mentions his name: (i) (In the first prologue of Proclus^) on the division of mathe- • Pappus, ed. Hultsch, p. 1026, 9. • Apollonius, ed. Heiberg, vol. ii. p. 170. ' Tannery, La GionUtrU grecque, pp. i8, 19. * Proclus, pp. 38, i — 41, 8. • Heron, ed. Hultsch, pp. 246, 16--249, '^• • Van Pesch, De Procti fontibus, pp. 97—113. The disserUtion of Tittel is entitled De Gemini Stoici studiis nuUhimaticU ( 1 895). 7 Proclus, pp. 38, I — 43, 8, except the allusion in p. 41, 8 — 10, to Ctesibius and Heron and 40 INTRODUCTION [ch. iv matical sciences into arithmctiCt geometry, mechanics, astronoiny, optics, geodesy, canonic (science of musical harmony^ and logistic (apparently arithmetical problems); (2) (in the note on the definition of a straight line) on the classification of lines (including curves) as simple (straight or circular) and mixed, composite and incomposite, uniform (o^oiofttp*!^) and non-uniform {avo^oiop€p€U\ lines "about solids" and lines produced by cutting solids, including conic and spiric sections' ; (3) (i^ the note on the definition of a plane surface) on similar distinctions extended to surfaces and solids* ; (4) (in the note on the definition of parallels) on lines which do not meet {acvpirrmTOi) but which are not on that account parallel, e,g a curve and its asymptote, showing that the property of not nueling does not make lines parallel — a favourite observation of Geminus — and^ incidentally, on bounded lines or those which mciose a figure and those which do not* ; (5) (in the same note) the definition of parallels given by Posidonius*; (6) on the distinction between postulates and axioms, the futility of trying to prove axioms, .as ApoUonius tried to prove Axiom i, and the equal incorrectness of assuming what really requires proof, "as Euclid did in the fourth postulate [equality of right angles] and in the fifth postulate [the parallel- postulate]^" ; (7) on Postulates i, 2, 3, which Geminus makes depend on the idea of a straight line being described by the motion of a point* ; (8) (in the note on Postulate 5) on the inadmissibility in geometry of an argument which is merely plausible, and the danger in this particular case owing to the existence of lines which do converge ad infiniium and yet never meet^ ; (9) (in the note on L i) on the subject-matter of geometry, theorems, problems and Stopt^fioi (conditions of possibility) for problems* ; (10) (in the note on I. 5) on a generalisation of I. 5 by Geminus through the substitution for the rectilineal base of ** one uniform line (curve)/' by means of which he proved that the only "uniform lines" their pneumatic devices {BavnarowoiXiHi), as regards which Proclus' authority may be Pappus (yni. p. 1094, 74 — ay) who uses very similar exjpressioiis. Heron, even if not later tnan ueminus, could hardly have been included in a historical work bv him. Perhaps Geminus may have referred to Ctesibius only, and Proclus may have inserted "and Heron*' himself. * Prochis, pp. 103, «i— 107, 10; pp. Ill, I— 113, 3. * tM, pp. riy, 14—110, IS, where perhaps in the passage pp. 117, 33 — ii8, 93 we may have Gemmus' own words. * iMd. pp. 176, 18 — 177, 95; perhaps also p. 175. The note ends with the words "These things too we have selected from Geminus* ^iXoiraXki for the elucidation of the matters in question.** Tannery ^p. 17) takes these words coming at the end of the commen- tary on the definitions as referring to the whole of the portion of the commentary dealing with the definitions. Van Pesch properly regards them as only applying to the note on paraiUis, This seems to me clear from the use of the word too (ro^'aOra ir aQ. * Proclus, p. 176, 5—17. * ihid, pp. 178— 18s, 4; pp. 183, 14— 184, 10; cf. p. 188, 3— II. * ibid. p. 185, 6 — 15. ' ibid. p. 193, 5 — 19. * iHd, pp. loo, ii — aoi, 35. CH. iv] PROCLUS AND HIS SOURCES 41 » , [ (alike in all their parts) are a straight line, a circle, and a cylindrical t; helix^ (11) (in the note on I. lo) on the question whether a line is made up of indivisible parts (dfiepr}), as affecting the problem of bisecting a given straight line* ; (12) (in the note on I. 35) on topical, or /<?rKJ-theorems*, where the illustration of the equal parallelograms described between a hyperbola and its asymptotes may also be due to Geminus*. Other passages which may fairly be attributed to Geminus, though his name is not mentioned, are the following : (i) in the prologue, where there is the same allusion as in the passage (8) above to a remark of Aristotle that it is equally absurd to expect scientific proofs from a rhetorician and to accept mere plausi- bilities from a geometer* ; (2) a passage in the prologue about the subject-matter, methods, and bases of geometry, the latter including axioms and postulates* ; (3) another on the definition and nature of eletnents^ ; (4) a remark on the Stoic use of the term axiom for every simple statement {air6<f>avai^ awXriY ; (5) another discussion on theorems and problems*, in the middle of which however there are some sentences by Proclus himself". (6) another passage, in connexion with Def. 3, on lines including or not including a figure (with which cf part of the passage (4) above)" ; (7) a classification of different sorts of angles according as they are contained by simple or mixed lines (or curves)"; (8) a similar classification of figures", and of plane figures" ; (9) Posidonius' definition of a figure^* ; (10) a classification of triangles into seven kinds" ; (11) a note distinguishing lines (or curves) producible indefinitely or not so producible, whether forming a figure or not forming a figure (like the "single-turn spiral")^'; (12) passages distinguishing different sorts of problems", different sorts of theorems", and two sorts of converses (complete and partial)*; (13) the definition of the term "porism" as used in the title of Euclid's Porisms, as distinct from the other meaning of "corollary ""; (14) a note on the Epicurean objection to I. 20 as being obvious even to an ass" ; (15) a passage on the properties of parallels, with allusions to • Proclus, p. 251, 2— II. • ibid. pp. 277, 25 — 370, 11. » ibid, pp. 394, 11—395, a and p. 395, 13—21. * ibid, p. 395, 8—12. • ibid. pp. 33, 21—34. I. • ibid. pp. 57, 9—58, 3. ' *^' PP- 72. 3— 75» 4- * if^' p. 77. 3—6- • ibid. pp. 77, 7—78, 13, and 79, 3—81, 4. " H>id. pp. 78, 13—79* «• ^» ibid. pp. 102, 31—103, '8. " ibid. pp. 116, 7—117, iC " ibid, pp. 159, II — 160, 9. " ibid, pp. 161, 37—164, 6. " ibid. p. 143, 5— II. " ibid. p. 168, 4—13. *' ibid. p. 187, 19—37. w 1^^. pp. 220, 7 — 333, 14; also p. 330, 6—9. " ibid, pp. 344, 14—346, 13. « ibid, pp. 353, 5—354, 30. » ibid. pp. 301, 31—303, 13. « ibid, pp. 333, 4— 3«3. 3- 4f INTRODUCTION [cw. iv Apollonms' C*?*kj, and the curves invented by Nicomedes, Hippias and Perseus' ; (t6) a passage on the parallel-postulate regarded as the converse of L ij\ Of the authors to whom Proclus was indebted in a less degree the most important is Apollonius of Perga. Two passages allude to his Ccni€s\ one to a work on irrationals*, and two to a treatise On /A# cocklias (apparently the cylindrical helix) by Apollonius*. But more important for our purpose are six references to Apollonius in connexion with elementary geometry, (i) He appears as the author of an attempt to explain the idea of a line (possessing length but no breadth) by reference to daily experience, e,g. when we tell someone to measure^ merely, the length of a road or of a wall*; and doubtless the similar passage showing how we may in like manner get a notion of a surface (without depth) is his also*. (2) He gave a new general definition of an angle*, . (3) He tried to prove certain axioms', and Proclus gives his I attempt to prove Axiom i, word for word"- J Proclus further quotes ; \ (4) Apollonius' solution of the problem in Eucl. L lo^ avoiding Euclid*s use of L 9", (5) his solution of the problem in l. 11, differing only slightly from Euclid's*', and (6) his solution of the problem in l. 23^. Heiberg" conjectures that Apollonius departed from Euclid's method in these propositions because he objected to solving problems of a more general, by means of problems of a more particular* character. Proclus however considers all three solutions inferior to Euclid's ; and his remarks on Apollonius' handling of these ele- mentary matters generally suggest that he was nettled by criticisms of Euclid in the work containing the things which he quotes from Apollonius, just as we conclude that Pappus was offended by the remarks of Apollonius about Euclid's incomplete treatment of the *• three- and four-line locus"." If this was the case, Proclus can hardly have got his information about these things at second-hand; and there seems to be no reason to doubt that he had the actual work of Apollonius before him. This work may have been the treatise mentioned by Marinus in the words ''Apollonius in his general treatise" (*A7roXXioi/io^ iv tJ icaOoKov irpayfjuireiay^ If the notice in the Fthrist^'' stating, on the authority of Thabit b. Qurra, that » Proclus, pp. 355, 10--356, 16. « ibid. p. 364. 9—12; pp. 3^4. ao— 365,4. » ibid. p. 71, lOjL-i*. 356, 8, 6. « ibid. p. 74, 23, 24. • ibid. pp. i&j, 5, 6, 14, 15. • ibid. p. 100, 5—19. ' ibid. p. 114, ao— 15. • ibid. p. 123, 15—19 (cf. p. 114, 17, p. 115, 17). • ibid. p. 183. 13, 14. w ibid, pp. 194. as— '95f 5« " ibid. pp. 179, 16—180, 4. '• ibid, p. 281, 8—19. " ibid. pp. 335, 16—336, 5. " PkihUgus, vol. XLliI. p. 489. ^ See above, pp. a, 3. >* Marinus in EucHdis Daia^ ed. Menge, p. S34, 16. » Fikrist^ tr. Sutcr, p. 19. i CH. IV] PROCLUS AND HIS SOURCES 43 Apollonius wrote a tract on the parallel-postulate be correct, it may have been included in the same work. We may conclude generally that, in it, Apollonius tried to remodel the beginnings of geometry, reducing the number of axioms, appealing, in his definitions of lines, surfaces etc., more to experience than to abstract reason, and substituting for certain proofs others of a more general character. The probabilities are that, in quoting from the tract of Ptolemy in which he tried to prove the parallel-postulate, Proclus had the actual work before him. For, after an allusion to it as **a certain book**' he gives two long extracts*, and at the beginning of the second indicates the title of the tract, '* in the (book) about the meeting of straight lines produced from (angles) less than two right angles," as he has very rarely done in other cases. Certain things from Posidonius are evidently quoted at second- hand, the authority being Geminus (e.g. the definitions of figtire and parallels) ; but besides these we have quotations from a separate work which he wrote to controvert Zeno of Sidon, an Epicurean who had sought to destroy the whole of geometry*. We are told that Zeno had argued that, even if we admit the fundamental principles (apxai) of geometry, the deductions from them cannot be proved without the admission of something else as well, which has not been included in the said principles*. On I. i Proclus gives at some length the argu- ments of Zeno and the reply of Posidonius as regards this proposition*. In this case Zeno's "something else" which he considers to be assumed is the fact that two straight lines cannot have a common segment, and then, as regards the "proof" of it by means of the bisection of a circle by its diameter, he objects that it has been assumed that two circumferences (arcs) of circles cannot have a common part Lastly, he makes up, for the purpose of attacking it, another supposed " proof" of the fact that two straight lines cannot have a common part. Proclus appears, more than once, to be quoting the actual words of Zeno and Posidonius ; in particular, two expres- sions used by Posidonius about "the acrid Epicurean" {jov Bpifiifv *EinKovp€iopy and his "misrepresentations" (Uoa-eiBwpti^ <fnfai rov Zi^vtDva avKo<f>avTelvy, It is not necessary to suppose that Proclus had the original work of Zeno before him, because Zeno's arguments may easily have been got from Posidonius' reply ; but he would appear to have quoted direct from the latter at all events. The work of Carpus mechanicus (a treatise on astronomy) quoted from by Proclus* must have been accessible to him at first-hand, because a portion of the extract from it about the relation of theorems and problems* is reproduced word for word. Moreover, if he were not using the book itself, Proclus would hardly be in a position to question whether the introduction of the subject of theorems and problems • Proclus, p. 191, 13. * ibid, pp. 361, 14— 3^3» ^8» PP- l^hy 7— 3^7» «7- • ibid, p. 100, 1 — 3. * ibid, pp. 199, 11 — 100, i. • ibid, pp. 314, 18 — 315, 13; pp. 116, 10 — 118, II. • ibid, p. 216, 21. ' ibid, p. 218, i. • ibid, pp. 341, 19 — 343, II. * ibid, pp. 143, 33—343, 11. L 44 INTRODUCTION [cH. iv was opportune in the place where it was found (et fiiv mari tcaipiv f} It is of course evident that Proclus had before him the original works of PlatOj Aristotle. Archimedes and Plotinus, as well as the Xvfifittcrd of Porphyry and the works of his master Syrianus (o i5^cTf/>o? ica0iiyffimpy, from whom he quotes in his note on the definition of an angle. Tannery also points out that he must have had before him a group of works representing the Pythagorean tradition on its mystic, as distinct from its mathematical, side, from Phitolaus downwards, and comprising the more or less apocryphal Upoi; \oyos of Pythagoras, the Oracles (Xojta), and Orphic verses". Besides quotations from writers whom wc can identify with more or less certainty, there are many other passages which are doubtless quoted from other commentators whose names we do not know. A list of such passages is given by van Pesch*, and there is no need to cite them here. Van Pesch also gives at the end of his work ' a convenient list of the books which, as the result of his investigation, he deems to have been accessible to and directly used by Proclus. The list is worth giving here, on the same ground of convenience. It is as follows: ^ Eudemus : history of giomttry. Gem in us : the theory of the mathematical sciejtces, r Heron : commmlary on the Elements of Euclid. Porphyry: „ „ „ . Pappus: „ „ „ j Apollonius of Perga : a work relating to elementary geometty* Ptolemy: on the parallel-postulate. Posidonius : a book controverting Zeno of Sidon. Carpus ; astronomy, Syrianus : a discussion on the angle. Pythagorean philosophical tradition, Plato s works » Aristotle's works. Archimedes' works. Plotinus: Enneades, Lastly we come to the question what passages, if any, in the commentary of Proclus represent his own contributions to the subject As we have seen, the onus probandi must be held to rest upon him who shall maintain that a particular note is original on the part of Proclus. Hence it is not enough that it should be impossible to point to another writer as the probable source of a note ; we must have a positive reason for attributing it to Proclus. The criterion must there- fore be found either (i) in the general terms in which Proclus points out the deficiencies in previous commentaries and indicates the respects in which his own will differ from them, or (2) in specific expressions used by him in introducing particular notes which may ^ Proclus, p. 341, 21, 33. ' ibid, p. 133, 19. ' Tannery, La Giomitrii gncque^ pp. 35, 36. * Van Pesch, Di Proclifontibus^ p. 139. • ibid. p. 155. u CH. iv] PROCLUS AND HIS SOURCES 45 indicate that he is giving his own views. Besides indicating that he paid more attention than his predecessors to questions requiring deeper study (ro irpayfunei^he^) and " pursued clear distinctions " (to evBiaiperop fji^rahiw/covra^y — by which he appears to imply that his predecessors had confused the different departments of their commentaries, viz. lemmas, cases, and objections {iparaaci^y — Proclus complains that the earlier commentators had failed to indicate the ultimate grounds or causes of propositions*. Although it is from Geminus that he borrowed a passage maintaining that it is one of the proper functions of geometry to inquire into causes (r^v airiav icaX TO tik riy, yet it is not likely that Geminus dealt with Euclid's propositions one by one ; and consequently, when we find Proclus, on \\ I. 8, 16, 17, 18, 32, and 47', endeavouring to explain causes, we have good reiEison to suppose that the explanations are his own. Again, his remarks on certain things which he quotes from Pappus can scarcely be due to anyone else, since Pappus is the latest of the commentators whose works he appears to have used. Under this head come (i) his objections to certain new axioms introduced by Pappus*, (2) his conjecture as to how Pappus came to think of his alterna- tive proof of I. 5^ (3) an addition to Pappus' remarks about the curvilineal angle which is equal to a right angle without being one*. The defence of Geminus against Carpus, who combated his view of theorems and problems, is also probably due to Proclus*, as well as an observation on I. 38 to the effect that I. 35 — 38 are really compre- hended in VI. I as particular cases^*. Lastly, we can have no hesitation in attributing to Proclus himself (i) the criticism of Ptolemy's attempt to prove the parallel-postulate >S and (2) the other attempted proof given in the same note^ (on I. 29) and assuming as an axiom that '' if from one point two straight lines forming an angle be produced ad infinitum the distance between them when so produced ad infinitum exceeds any finite magnitude (i.e. length)," an assumption which purports to be the equivalent of a statement in Aristotle". It is introduced by words in which the writer appears to claim originality for his proof: "To him who desires to see this proved {KaTaa'tc€va^6fi€vop) let it be said by us (Keyiado) irap ^fjLoSv)" etc." Moreover, Philoponus, in a note on Aristotle's Anal, post, I. 10, says that **the geometer (Euclid) assumes this as an axiom, but it wants a great deal of proof, insomuch that both Ptolemy and Proclus wrote a whole book upon it"." > Proclus, p. 84, 13, p. 432, 14, • ibid, - -- -- CHAPTER V. THE TEXT", It is well known that the title of Simson's edition of Euclid (first brought out in Latin and English in 1756) claims that, in it, "the errors by which Theon, or others, have long ago vitiated these books are corrected, and some of Euclid's demonstrations arc restored " ; and readers of Simson's notes are familiar with the phrase^ used, where anything in the text does not seem to htm satisfactory, to the effect that the demonstration has been spoiled, or things have been interpo- lated or omitted, by Theon "or some other unskilful editor/' Now most of the MSS. of the Greek text prove by their titles that they proceed from the recension of the Elements by Theon ; they purport to be either " from the edition of Theon " [ix t^9 ^iti^vo^ €xSo<r€<a?) or ** from the lectures of Theon *' (agro avvovaimu rov %itapo^). This was Theon of Alexandria (4th c. A.D,) who also wrote a commentary on Ptolemy, in which there occurs a passage of the greatest importance in this connexion*: "But that sectors in equal circles are to one another as the angles on which they stand Ans been proved by me in my edition of the Elements at tfu end of the sixth book!' Thus Theon himself says that he edited the Elements and also that the second part of VI. 33, found in nearly all the MSS,, is his addition. This passage is the key to the whole question of Theon*s changes in the text of Euclid ; for, when Peyrard found in the Vatican the MS. 190 which contained neither the words from the titles of the other MSS. quoted above nor the interpolated second part of VI. 33, he was justified in concluding, as he did, that in the Vatican MS, we have an edition more ancient than Theon 's. It is also clear that the copyist of P,or rather of its archetype, had before him the two recensions and systematically gave the preference to the earlier one ; for at XIII. 6 in P the first hand has added a note in the margin 1 ** This theorem is not given in most copies of the new editimt, but is found in those of the old," Thus we are more fortunate than Simson, since our judgment of Theon*s recension can be formed on the basis, not of mere conjecture, but of the documentary evidence afforded by a comparison of the Vatican MS. just mentioned with what we may conveniently call, after Heiberg, the Theonine MSS* * The materkl for the whole af this chapter is Uken firom Heiberg's cdtticm of the EUmenii^ introduction to vol. V., And itotn tbe same schoUr's Uiierargtsatuhiiuk^ SittdUn iiber Euklidy p. 174 sqq. ^d Fi^ralipomtna fv Euklidtn Htrma^ XXXV I It.^ 1905. * I. p. aoj ed. Hainia = p. 50 etL }^iSA\. CH. v] THE TEXT 47 The MSS. used for Heibei^s edition of the Elements are the following : (i) P = Vatican MS. numbered 190, 4to, in two volumes (doubt- less one originally) ; loth c. This is the MS. which Peyrard was able to use ; it was sent from Rome to Paris for his use and bears the stamp of the Paris Imperial Library on the last page. It is well and carefully written. There are corrections some of which are by the original hand, but generally in paler ink, others, still pretty old, by several different hands, or by one hand with different ink in different places (P m. 2), and others again by the latest hand (P m. rec). It contains, first, the Elements I. — XIII. with scholia, then Marinus' commentary on the Data (without the name of the author), followed by the Data itself and scholia, then the Elements XIV., XV. (so called), and lastly three books and a part of a fourth of a commentary by Theon €t9 toi)? irpox^ipov^ icavova^ OtoXc- liaiov. The other MSS. are " Theonine." (2) F = MS. XXVIII, 3, in the Laurentian Library at Florence, 4to ; loth c. This MS. is written in a beautiful and scholarly hand and contains the Elements I. — XV., the Optics and the Phaenomena, but is not well preserved. Not only is the original writing renewed in many places, where it had become faint, by a later hand of the i6th c, but the same hand has filled certain smaller lacunae by gumming on to torn pages new pieces of parchment, and has replaced bodily certain portions of the MS., which had doubtless become iil^ible, by fresh leaves. The larger gaps so made good extend from Eucl. vii. 12 to IX. 1 5, and from XII. 3 to the end ; so that, besides the conclusion of the Elements, the Optics and Phaenomena are also in the later hand, and we cannot even tell what in addition to the Elements I. — XIII. the original MS. contained. Heiberg denotes the later hand by ^ and observes that, while in restoring words which had become faint and filling up minor lacunae the writer used no other MS., yet in the two larger restorations he used the Laurentian MS. xxviil, 6, belonging to the 13th — 14th c. The latter MS. (which Heiberg denotes by f) was copied from the Viennese MS. (V) to be described below. (3) B = Bodleian MS., D'Orville X. i inf. 2, 30, 4to ; A.D. 888. This MS. contains the Elements I. — xv. with many scholia. Leaves 15 — 118 contain L 14 (from about the middle of the proposition) to the end of Book VI., and leaves 123 — 387 (wrongly numbered 397) Books VII. — XV. in one and the same elegant hand (9th c). The leaves preceding leaf 1 5 seem to have been lost at some time, leaves ' 6 to 14 (containing Elem, I. to the place in I. 14 above referred to) being carelessly written by a later hand on thick and common parch- ment (13th c). On leaves 2 to 4 and 122 are certain notes in the hand of Arethas, who also wrote a two-line epigram on leaf 5, the greater part of the scholia in uncial letters, a few notes and corrections, and two sentences on the last leaf, the first of which states that the MS. was written by one Stephen clericus in the year of the world 6397 { 48 INTRODUCTION [cm. v (^ 888 A.D,)j while the second records Arethas* own acqiiisition of it Arethas lived from, say, 865 to 939 A.D. He was Archbishop of Caesarea and wrote a commentary on the Apocalypse, The portions of his library which survive are of the greatest interest to palaeography on account of his exact notes of dates, names of copyists, prices of parchment etc. It is to him also that we owe the famous Plato MS. from Pat m OS (Cod. Clarkianus) which was written for him in November 895 », (4) V = Viennese MS. Philos, Gn No. 103 ; probably 12th c This MS* contains 292 leaves. Eucl Ekments h — XV. occupying leaves 1 to 254, after which come the Optics (to leaf 271), the PAacn(^fn€fta {mxyWlTLX^d at the end) from leaf 272 to leaf 2S2, and lastly scholia^ on leaves 283 to 292^ also imperfect at the end. The different materia] used for different parts and the varieties of handwriting make it necessary for Heiberg to discuss this MS, at some length*. The handwriting on leaves 1 to 183 (Book h to the middle of X. 105) and on leaves 203 to 234 (from XL 31, towards the end of the proposition, to XIII. 7, a few lines down) is the same ; between leaves 184 and 202 there arc two varieties of handwriting, that of leaves 184 to '189 and that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in the same handwriting, changes first gradually into that of leaves 184 to 1 89 and then (verso) into a third more rapid cursive writing which is the same as that of the greater part of the scholia, and also as that of leaves 243 and 282, althoughj as these leaves are of different material, the look of the writing and of the ink seems altered There are corrections both by the first and a second hand, and scholia by many hands. On the whole, in spite of the apparent diversity of handwriting in the MS., it is probable that the whole of it was written at about the same time, and it may (allowing for changes of material, ink etc) even have been written by the same man. It is at least certain that, when the Laurentian M.s* xxviii^O was copied from it, the whole MS. was in the condition in which it is now, except as regards the later scholia and leaves 283 to 292 which are not in the laurentian MS.j that MS. coming to an end where the F/taetwrnena breaks off abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c. But it was apparently in two volumes originally, the first-con- sisting of leaves i to 183 ; and it is certain that it was not all copied at the same time or from one and the same original. For leaves 1 84 to 202 were evidently copied from two MSS. different both from one another and from that from which the rest was copied. Leaves 1 84 to the middle of leaf 1 89 (recto) must have been copied from a MS, similar to P^ as is proved by similarity of readings, though not from P itself The rest, up to leaf 202, were copied from the Bolc^na MS, (b) to be mentioned below. It seems clear that the content of leaves 184 to 202 was supplied from other MSS. because there was a lacuna in the original from which the rest of V was copied ] ] < \ ^ See Pauly-WUsowa, Rial-EncydopadU der class. AlUrtumtunssensckaftiHt voL IL, 18961 eibeig, vol v. pp. zxix— xxxiii. p. 67^^ r' cav] THE TEXT 49 Heiberg sums up his conclusions thus. The copyist of V first copied leaves i to 183 from an original in which two quaterniones were missing (covering from the middle of Eucl. x. 105 to near the end of XL 31). Noticing the lacuna he put aside one quatemio of the parchment used up to that point Then he copied onwards from the end of the lacuna in the original to the end of the Phamomena, After this he looked about him for another MS. from which to fill up the lacuna ; finding one, he copied from it as far as the middle of leaf 189 (recto). Then, noticing that the MS. from which he was copying was of a different class, he had recourse to yet another MS. from which he copied up to leaf 202. At the sanre time, finding that the lacuna was longer than he had reckoned for, he had to use twelve more leaves of a different parchment in addition to the quatemio which he had put aside. The whole MS. at first formed two volumes (the first containing leaves i to 183 and the second leaves 184 to 282); then, after the last leaf had perished, the two volumes were made into one to which two more quaterniones were also added. A few leaves of the latter of these two have since perished. (5) b s MS. numbered 18 — 19 in tlie Communal Library at Bologna, in two volumes, 4to; nth c. This MS. has scholia in the margin written both by the first hand and by two or three later hands ; some are written by the latest hand, Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 14th c) who owned the MS. at one time. It contains {a) in 14 quater- niones the definitions and the enunciations (without proofs) of the Elements I. — XIIL and of the Data^ {b) in the remainder of the volumes the Proem to Geometry (published among the Variae Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) followed by the Elements I.— XIII. (part of XIII. 18 to the end being missing), and then by part of the Data (from the last three words of the enunciation of Prop. 38 to the end of the penultimate clause in Prop. 87, ed. Menge). From XI. 36 inclusive to the end of xil. this MS. appears to represent an entirely different recension. Heiberg is compelled to give this portion of b separately in an appendix. He conjectures that it is due to a Byzantine mathematician who thought j- Euclid's proofs too long and tiresome and consequently contented 11 himself with indicating &e course followed^ At the same time this Byzantine must have had an excellent MS. before him, probably of the . ante-Theonine variety of which the Vatican MS. 190 (P) is the sole representative. (6) p = Paris MS. 2466, 4to; 12th c. This manuscript is written in two hands, the finer hand occupying leaves i to 53 (recto), and a more careless hand leaves 53 (verso) to 64, which are of the same parchment as the earlier leaves, and leaves I 65 to 239, which are of a thinner and rougher parchment showing traces of writing of the 8th — 9th c. (a Greek version of the Old Testament). The MS. contains the Elements I. — xill. and some scholia after Books xi., XII. and Xlll. > Ztiischrifi fiir Math, u. Physik^ XXIX., hist.-litt. Abtheilung, p. 13. H. E. 4 I- 50 INTRODUCTION [cK, v (7) q = Pans MS. 2344, folio ; 12th c It is written by one hand but includes scholia by many hands. On leaves l to 16 (recto) are scholia with the same title as that found by Wachsmuth in a Vatican MS. and relied upon by him to prove that Proclus continued his commentaries beyond Book L' Leaves 17 to 357 contain the Elements L — XIIL (except that there is a lacuna from the middle of VIIL 25 to the Sxffetrt*^ of IX, 14); before Books VIL and X, there are some leaves filled with scholia only* and leaves 358 to 366 contain nothing but scholia, (8) Heiberg also used a palimpsest in the British Museum (Add, 1 72 11). Five pages are of the 7th — 8th c, and are contained (leaves 49 — S3) in the second volume of the Syrian ms. Brit. Mus. 687 of the 9th c, ; half of leaf 50 has perished » The leaves contain various frag- ments from Book X. enumerated by Heiberg, Vol. III., p, v, and nearly the whole of X II I, 14, Since his edition of the Elements was published, Heiberg has collected further material bearing on the history of the text*. Besides giving the results of further or new examination of MSS,. he has collected the fresh evidence contained in an-Nairizl's comment a r>\ and particularly in the quotations from Heron *3 commentary given in it (often word for word^ which enable us in several cases to trace I differences between our text and the text as Heron had it, and to identify some interpolations which actually found their way into the text from Heron's commentary itself; and lastly he has dealt with some valuable fragments of ancient papyri which have recently come to light, and which are especially important in that the evidence drawn from them necessitates some modification in the views expressed in the preface to VoL V, as to the nature of the changes made in Theon's recension, and in the principles laid down for differentiating between Theon's recension and the original text, on the basis of a comparison between P and the Theonine MSS. alone* The fragments of ancient papyri referred to are the following, I* Pafyrus Herculanensis No. 1061* This fragment quotes Def. i j of Book l. in Greek, and omits the words ^ ^aX^trat 7r€pi<f>ip€ta^ "which is called the circumference/' found in all our MSS., and the further addition ^/>ot rtju roi) /WKXav TTipitf^tp^tav also found in practically all the MSS. Thus Heibei^'s assumption that both expressions are interpolations is now confirmed by this oldest of all sources, 2. T^ Oxyrhymhus Papyri 1. p. 58, No. XXIX* of the 3rd or 4th c. This fragment contains the enunciation of Eucl IL 5 (with figure, apparently without letters, immediately following, and not, as usual in our MSS., at the end of the proof) and before it the part of a word"" vepiexofie belonging to II. 4 (with room for -yf» opOo^vUp* iirep l^i ^ [tit rjk T9O EAxXcldov rrocxcM wpdXa^ifiaifi/uwa 4k rQif np6ffXov aw^fdimif mi car* fri- n^^. Cf p. 3a, note 8, above. ' Heibeig, Faratipomtna mm EukHd in Hermes^ xxxviiu, 15^3, pp. 46 — 74, 161 — aoi, ' Described by Heibeig in Otfersigt aver dit kmgi. damJki VuUmkabemis Siiskttbs Forhandtingpr^ 1900, p. 161. . . (" CH.V] THE TEXT 51 Set^ai and a stroke to mark the end), showing that the fragment had not the Porism which appears in all the Theonine MSS. and (in a later hand) in P, and thereby confirming Heiberg's assumption that the Porism was due to Theon. 3. A fragment in Fayum towns and ttieir papyri^ p. 96, No. IX. of 2nd or 3rd c This contains I. 39 and I. 41 following one another and almost complete, showing that I. 40 was wanting, whereas it is found in all the MSS. and is recognised by Proclus. Moreover the text of the beginning of I. 39 is better than ours, since it has no double hiopur/AS^ but omits the first (" I say that they are also in the same parallels ") and has " and** instead of **/or let ^Z> be joined " in the next sentence. It is clear that I. 40 was interpolated by someone who thought there ought to be a proposition following I. 39 and related to it as I. 38 is related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses I. 40, and therefore was not likely to include it The same interpolator failed to realise that the words "let AD he joined" were part of the i/cOeai^ or setting-out, and took them for the tcaraa-tcewj or " construc- tion " which generally follows the hiopurfio^ or " particular statement " of the conclusion to be proved, and consequently thought it necessary to insert a Siopurfio^ before the words. The conclusions drawn by Heiberg from a consideration of particular readings in this papyrus along with those of our MSS. will be referred to below. We now come to the principles which Heiberg followed, when preparing his edition, in differentiating the original text from the Theonine recension by means of a comparison of the readings of P and of the Theonine MSS. The rules which he gives are subject to a certain number of exceptions (mostly in cases where one Ms. or the other shows readings due to copyists' errors), but in general they may be relied upon to give conclusive results'. The possible alternatives which the comparison of P with the Theonine MSS. may give in particular passages are as follows : I. There may be agreement in three different degrees. (i) P and all the Theonine MSS. may agree. In this case the reading common to all, even if it is corrupt or interpolated, is more ancient than Theon, i.e. than the 4th c. (2) P may agree with some (only) of the Theonine MSS. In this case Heiberg considered that the latter give the true reading of Theon's recension, and the other Theonine MSS. have departed from it (3) P and one only of the Theonine MSS. may agree. In this case too Heiberg assumed that the one Theonine MS. which agrees with P gives the true Theonine reading, and that this rule even supplies a sort of measure of the quality and faithfulness of the Theonine MSS. Now none of them agrees alone with P in preserving the true reading so often as F. Hence F must be held to have pre- served Theon 's recension more faithfully than the other Theonine MSS.; and it would follow that in those portions where F fails us P must 4—2 L $2 INTRODUCTION [ch. cany rather more weight even though it may differ from the Theonine MSS. BVpq. (Heiberg gives many examples en proof of this, as of his main rules generally, for which reference must be made to his Prole- gomena in Vol V.) The specially close relation of F and P is also illustrated by passages in which they have the same errors \ the explanation of these common errors (where not due to accident) is found by Heiberg in the supposition that they existed, but were not noticed by Theon, in the original copy in which he made his changes. Although however F is by far the best of the Theonine MSS., there are a considerable number of passages where one of the others (B, V, p or q) alone with P gives the genuine reading of Theon's recension. As the result of the discovery of the papyrus fragment containing I. 39, 41, the principles above enunciated under (2) and (3) are found by Heiberg to require some qualification. For there is in some cases a remarkable agreement between the papyrus and the Theonine MSS. (some or all) as against R This shows that Theon took more trouble to follow older MSS,, and made fewer arbitrary changes of his own, than has hitherto been supposed. Next, when the papyrus agrees with some of the Theonine MSS. against P, it must now be held that these MSS. (and not, as formerly supposed, those which agree with P) give the true reading of Theon. If it were otherwise, the agreement between the papyrus and the Theonine MSS. would be accidental: but it happens too often for this. It is clear also that there must have been contamination between the two recensions ; otherwise, whence could the Theonine MSS, which agree with P and not with the papyrus have got their readings? The influence of the P class on the Theonine F is especially marked. 11. There may be disagreement between P and all the Theonine MSS. The following possibilities arise. (1) The Theonine MSS. differ also among themselves* In this case Heiberg considered that P nearly always has the true reading, and the Theonine MSS, have suffered interpolation in different ways after Theon's time. (2) The Theonine Mss. all combine against P. In this case the explanation was assumed by Heiberg to be one or other of the following. (a) The common reading is due to an error which cannot be imputed to Theon (though it may have escaped him when putting together the archetype of his edition); such error may either have arisen accidentally in all alike, or (more frequently) may be referred to a common archetype of all the MSS. (/3) There may be an accidental error in P ; e.g. something has dropped out of P in a good many places, generally through (<y) There may be words interpolated in P. (0) Lastly, we may have in the Theonine MSS. a change made by Theon himself, (The discovery of the ancient papyrus showing readings agreeing i CH. v] THE TEXT 53 with some, or with all, of the Theonine MSS. against P now makes it necessary to be very cautious in applying these criteria.) It is of course the last class (8) of changes which we have to investigate in order to get a proper idea of Theon's recension. Heiberg first observes, as regards these, that we shall find that Theon, in editing the Elements, altered hardly anything without some reason, often inadequate according to our ideas, but still some reason which seemed to him sufficient Hence, in cases of very slight differ- ences where both the Theonine MSS. and P have readings good and probable in themselves, Heiberg is not prepared to put the differences down to Theon. In those passages where we cannot see the least reason why Theon, if he had the reading of P before him, should have altered it, Heiberg would not at once assume the superiority of P unless there was such a consistency in the differences as would indicate that they were due not to accident but to design. In the absence of such indications, he thinks that the ordinary principles of criticism should be followed and that proper weight should be attached to the antiquity of the sources. And it cannot be denied that the sources of the Theonine version are the more ancient. For not only is the British Museum palimpsest (L), which is intimately connected with the rest of our MSS., at least two centuries older than P, but the other Theonine MSS. are so nearly allied that they must be held to have had a common archetype intermediate between them and the actual edition of Theon ; and, since they themselves are as old as, or older than P, their archetype must have been much older. Heiberg gives (pp. xlvi, xlvii) a list of passages where, for this reason, he has followed the Theonine MSS. in preference to P. It has been mentioned above that the copyist of P or rather of its archetype wished to give an ancient recension. Therefore (apart from clerical errors and interpolations) the first hand in P may be relied upon as giving a genuine reading even where a correction by the first hand has been made at the same time. But in many places the first hand has made corrections afterwards ; on these occasions he must have used new sources, e.g. when inserting the scholia to the first Book which P alone has, and in a number of passages he has made additions from Theonine MSS. We cannot make out any " family tree " for the different Theonine MSS. Although they all proceeded from a common archetype later than the edition of Theon itself, they cannot have been copied one from the other ; for, if they had been, how could it have come about that in one place or other each of them agrees alone with P in pre- serving the genuine reading } Moreover the great variety in their agreements and disagreements indicates that they have all diverged to about the same extent from their archetype. As we have seen that P contains corrections from the Theonine family, so they show correc- tions from P or other MSS. of the same family. Thus V has part of the lacuna in the MS. from which it was copied filled up from a MS. similar to P, and has corrections apparently derived from the same ; the copyist, however, in correcting V, also used another MS. to which 1 54 INTRODUCTION [ch, v he alludes in the additions to ix, 19 and 30 (and also on X, 23 For); "in the book of the Ephesian (this) is not found*' Who this Ephesian of the 1 2th c was, we do not know. We now come to the alterations made by Theon in his edition of the Elements. I shall indicate classes into which these alterations may be divided but without details (except in cases where they affect the mathematical contmt as distinct from form or language pure and simple)*. I. Alterations made by Theon where he founds or thought kt founds mistakes in the original 1. Real blots in the original which Theon saw and tried to remove. {a) Euclid has a porism (corollary) to vi. 19, the enunciation of which speaks of similar and similarly described figures though the proposition itself refers only to triangles, and therefore the porism should have come after VI. 20. Theon substitutes triangle for figtirt and proves the inore general porism after VL 20. Qf) In IX. 19 there is a statement which is obviously incorrect. Theon saw this and altered the proof by reducing four alternatives to two, with the result that it fails to correspond to the enunciation even with Theon's substitution of** if*' for '* when " in the enunciation, {c) Theon omits a porism to IX* 11, although it is necessary for the proof of the succeeding proposition, apparently because, owing to an error in the text {Kmrk ^qv corrected by Heiberg into hti th)^ he could not get out of it the right sense. (rf) I should also put into this categoiy a case which Heiberg classifies among those in which Th^:^n merely fancied that he found mistakes, viz. the porism to V. 7 stating that, if four magnitudes are proportional, they are proportional inversely. Theon puts this after V. 4 with a proof, which however has no necessary connexion with V. 4 but is obvious from the definition of proportion, {f) I should also put under this head Xl. i, where Euclid s argu- ment to prove that two straight lines cannot have a common segment is altered. 2. Passages which seemed to Theon to contain blots, and which he therefore set himself to correct, though more careful consideration would have shown that Euclid's words are right or at least may be excused and offer no difficulty to an intelligent reader. Under this head come : (a) an alteration in ill. 24. \b) a perfectly unnecessary alteration, in vi. 14, of "equiangular parallelograms" into "parallelograms having one angle equal to one angle,'* where Theon followed the false analogy of vi. 1 5. {c) an omission of words in v. 26, owing to his having been mis- led by a wrong figure. (rf) an alteration of the order of XI. Deff. 27, 28. \e) the substitution of " parallelepipedal solid " for " cube *' in XI. ^ Exhaustive detaUs mider aU the difleient heads are given by Heibeig (VoL v. pp. lii— bov). CH. v] THE TEXT 55 38, because Theon observed, correctly enough, that it was true of the parallelepipedal solid in general as well as of the cube, but failed to give weight to the fact that Euclid must have given the particular case of the cube for the simple reason that that was all he wanted for use in xill. 17. (/) the substitution of the letter ^ for ft ( F for Z in my figure) because he saw that the perpendicular from K to B^ would fall on ^ itself, so that ^, ft coincide. But, if the substitution is made, it should ht proved that 4>, ft coincide. Euclid can hardly have failed to notice the fact, but it may be that he deliberately ignored it as unnecessary for his purpose, because he did not want to lengthen his proposition by giving the proof II. Emendations intended to improve the form or diction of Euclid. Some of these emendations of Theon affect passages of appreciable length. Heiberg notes about ten such passages ; the longest is ^ in Eucl. XII. 4 where a whole page of Heiberg's text is affected and I Theon's version is put in the Appendix. The kind of alteration may be illustrated by that in ix. 15 where Euclid uses successively the propositions Vll. 24, 25, quoting the enunciation of the former but not of the latter ; Theon does exactly the reverse. In a few of the cases here quoted by Heiberg, Theon shortened the original somewhat. But, as a rule, the emendations affect only a few words in each sentence. Sometimes they are considerable enough to alter the con- formation of the sentence, sometimes they are trifling alterations "more magistellorum ineptorum" and unworthy of Theon. Generally speaking, they were prompted by a desire to change anything which was out of the common in expression or in form, in order to reduce the language to one and the same standard or norm. Thus Theon changed the order of words, substituted one word for another where the latter was used in a sense unusual with Euclid (e.g. iweiiiprcp, "since," for Sn in the sense of "because"), or one expression for another in like circumstances (e.g. where, finding ''that which was enjoined would be done" in a theorem, Vll. 31, and deeming the phrase more appropriate to a problem, he substituted for it " that which is sought would be manifest"; probably also and for similar reasons he made certain variations between the two expressions usual at the end of propositions oirtp Ihei Sel^ai and Swep eSet iroifjaai, quod erat demonstrandum and quod erat faciendum). Sometimes his alterations show carelessness in the use of technical terms, as when he uses iiTTcaOai (to meet) for i^dtrr^cBai (to touch) although the ancients carefully distinguished the two words. The desire of keeping to a standard phraseology also led Theon to omit or add words in a number of cases, and also, sometimes, to change the lettering of figures. But Theon seems, in editing the Elements, to have bestowed the most attention upon III. Additions designed to supplement or explain Euclid, First, he did not hesitate to interpolate whole propositions where be thought there was room or use for them. We have already 56 INTRODUCTION mentioned the addition to vi. 33 of the second part relating to sectors^ for which Theon himself takes credit in his commentary on Ptolemy. Again, he interpolated the proposition commoniy known as VI L 22 {ex aequo in propartione perturbata for numbers, corresponding to V, 23X and perhaps also vil 20, a particular case of viL 19 as VL 17 is of VL 16. He added a second case to VL 27, a porism to IL 4, a second porism to III. 16, and a lemma after X. 12; perhaps also the porism to V. 19 and the first porism to vi, 2a He also inserted alternative proofs here and there, e.g, in IL 4 (where the alternative differs little from the original) and in vn. 31 ; perhaps also in X. i, 6, and 9* Secondly, he sometimes repeats an argument where Euclid had said "For the same reason/* adds specific references to points, straight lines etc. in the figures in order to exclude the possibility of mistake arising from Euclid's reference to them in general terms, or inserts words to make the meaning of Euclid more plain, e.g. componendo and alternately^ where Euclid had left them out. Some- times he thought to increase by his additions the mathematical precision of Euclid^s language in enunciations or elsewhere, sometimes to make smoother and clearer things which Euclid had expressed with unusual brevity and harshness or carelessness, in reliance on the intelligence of his readers* Thirdly, he supplied intermediate steps where EucHd^s argument seemed too rapid and not easy enough to follow. The form of these additions varies; they are sometimes placed as a definite intermediate step with "therefore*' or "so that" sometimes they are additions to the statement of premisses, sometimes phrases introduced by ** since " "for" and the like, after the inference. Lastly, there Is a very large class of additions of a word, or one or two words, for the sake of clearness or consistency. Heiberg gives a number of examples of the addition of such nouns as "triangle," "square," ''rectangle,** "magnitude/* "number" "pointy" "side," "circle," "straight line" "area" and the like, of adjectives such as " remaining/' " right," " whole/' " proportional/' and of other parts of speech, even down to words like " is " {i<irl) which is added 600 times. Si;, &pa^ ^ikp, ydp, scat and the like. IV. Omissions by Tksan. Heibei^ remarks that, Theon *s object having been, as above shown, to amplify and explain Euclid, we should not naturally have expected to find him doing much in the contrary process of com- pression, and it is only owing to the recurrence of a certain sort of omissions so frequently (especially in the first Books) as to exclude the hypothesis of their being all due to chance that we are bound to credit him with alterations making for greater brevity. We have seen, it is true, that he made omissions as well as additions for the purpose of reducing the language to a certain standard form. But there are also a good number of cases where in the enunciation of propositions, and in the exposition (the re-statement of them with reference to the figure), he has left out words because, apparently, he regarded Euclid's language as being too careful and precise. CH.V] THE TEXT 57 Again, he is apparently responsible for the frequent omission of the words iir^p ioei Bel^i (or 'n'oi,fiaa4)t Q.E.D. (or F.), at the end of propositions. This is often the case at the end of porisms, where, in omitting the words, Theon seems to have deliberately departed from Euclid's practice. The MS. P seems to show clearly that, where Euclid put a porism at the end of a proposition, he omitted the Q.E.D. at the end of the proposition but inserted it at the end of the porism, as if he regarded die latter as being actually a part of the proposition itself. As in the Theonine MSS. the Q.E.D. is generally omitted, the omission would seem to have been due to Theon. Sometimes in these cases the Q.E.D. is interpolated at the end of the proposition. Heiberg summed up the discussion of Theon's edition by the remark that Theon evidently took no pains to discover and restore from MSS. the actual words which Euclid had written, but aimed much more at removing difficulties that might be felt by learners in studying the book. His edition is therefore not to be compared with the editions of the Alexandrine grammarians, but rather with the work done by Eutocius in editing Apollonius and with an interpolated recension of some of the works of Archimedes by a certain Byzantine, Theon occupying a position midway between these two editors, being superior to the latter in mathematical knowledge but behind Eutocius in industry (these views now require to be some- what modified, as above stated). But however little Theon's object may be approved by those of us who would rather know the ipsissima verba of Euclid, there is no doubt that his work was approved by his pupils at Alexandria for whom it was written ; and his edition was almost exclusively used by later Greeks, with the result that the more ancient text is only preserved to us in one MS. As the result of the above investigation, we may feel satisfied that, where P and the Theonine MSS. agree, they give us (except in a few accidental instances) Euclid as he was read by the Greeks of the 4th c. But even at that time the text had been passed from hand to hand through more than six centuries, so that it is certain that it had already suffered changes, due partly to the fault of copyists and partly to the interpolations of mathematicians. Some errors of copyists escaped Theon and were corrected in some MSS. by later hands. Others appear in all our MSS. and, as they cannot have arisen accidentally in all, we must put them down to a common source more ancient than Theon. A somewhat serious instance is to be found in III. 8 ; and the use of amriadta for i^irriaOta in the sense of "touch" may also be mentioned, the proper distinction between the words having been ignored as it was by Theon also. But there are a number of imperfections in the ante-Theonine text which it would be unsafe to put down to the errors of copyists, those namely where the good MSS. agree and it is not possible to see any motive that a copyist could have had for altering a correct reading. In these cases it is possible that the imperfections are due to a certain degree of carelessness on the part of Euclid himself; for it S8 INTRODUCTION [ch. v is not p(^sib1e ** Euclidem ab omni naevo vindicare/' to use the words of SaccheriSand consequently Simson ts not right in attributing to Theon and other editors all the things in EucHd to which mathe- matical objection can be taken. Thiis^ when EucHd speaks of "the ratio compounded of the sides" for *'the ratio compounded of the ratios of the sides/* there is no reason for doubting that Euclid himself is responsible for the more slip-shod expression. Again, in the Books XI. — XIIL relating to solid geometry there arc blots neither few nor altc^ether unimportant which can only be attributed to Euclid himself*; and there is the less reason for hesitation in so attributing them because solid geometry was then being treated in a thoroughly systematic manner for the first time. Sometimes the cofulusion {cvfvrripcLciia) of a proposition does not correspond exactly to the enunciation, often it is cut short with the w^ords koX ra efij? " and the rest" (especially from Book X. onwards), and very often in Books VIII*, IX. it is omitted. Where all the MSS, agree, there is no ground for hesitating to attribute the abbreviation or omission to Euclid ; though, of course, where one or more Ms*>. have the longer form, it must be retained because this is one of the cases where a copyist has a temptation to abbreviate. Where the true reading is preserved in one of the Theonine MSS, alone, Heibcrg attributes the wrong reading to a mistake which arose before Theon's time, and the right reading of the single MS, to a successful correction* We now come to the most important question of the Interpolations introduced befare Tkeatis time. I. Alternative proofs or additional cases* It is not in itself probable that Euclid would have given two proofs of the same proposition ; and the doubt as to the genuineness of the alternatives is increased when we consider the character of some of them and the way in which they are introduced. First of all, we have those of VL 20 and xn. 17 introduced by '*we shall prove this otherwise more readily (wpox^ipojef^opy* or that of X, 90 " it is possible to prove mare slwrtly (<yvifrofiwr€popy Now it is impossible to suppose that Euclid would have given one proof as that definitely accepted by him and then added another with the express comment that the latter has certain advantages over the former. Had he con- sidered the two proofs and come to this conclusion, he would have inserted the latter in the received text instead of the former. These alternative proofs must therefore have been interpolated. The same argimient applies to alternatives introduced with the words " or even thus" (fj «cai o{hrfl09)> ''or even otherwise" (tf teal aXXo^). Under this head come the alternatives for the last portions of IIL 7, 8 ; and Heibcrg also compares the alternatives for parts of IIL 31 (that the angle in a semicircle is a right angle) and xill. 18, and the alternative proof of the lemma after x. 32. The alternatives to x. 105 and 106, ^ EmcUda ab cmm uoivo vindUatus^ Mediolani, T733. * Cf. espediUy the anomption, without proof or definition, of the criterion for eqtml solid aa^et, and the incomplete proof of xn. 17. I CH. V f aeaii CH. v] THE TEXT 59 again, are condemned by the place in which they occur, namely after an alternative proof to X. 115. The above alternatives being all admitted to be spurious, suspicion must necessarily attach to the few others which are in themselves unobjectionable. Heibei^ instances the alternative proofs to III. 9, III. 10, VI. 30, VI. 3 1 and XI. 22, observing that it is quite comprehensible that any of these might have occurred to a teacher or editor and seemed to him, rightly or wrongly, to be better than the corresponding proofs in Euclid. Curiously enough, Simson adopted the alternatives to HI. 9, 10 in preference to the genuine proofs. Since Heiberg's preface was written, his suspicion has been amply confirmed as regards III. 10 by the commentary of an-NairlzI (ed. Curtze) which shows not only that this alternative is Heron's, but also that the substantive proposition III. 12 in Euclid is also Heron's, having been given by him to supplement III. 11 which must originally have been enunciated of circles " touching one another" simply, i.e. so as to include the case of external as well as internal contact, though the proof covered the case of internal contact only. "Euclid, in the iith proposition," says Heron, "supposed two circles touching one another internally and wrote the proposition on this case, proving what it was required to prove in it But I will show haw it is to be proved if the contact be extemalK** This additional proposition of Heron's is by way of adding another case^ which brings us to that class of interpolation. It was the practice of Euclid and the ancients to give only one case (generally the most difficult one) and to leave the others to be investigated by the reader for himself. One interpolation of a second case (vi. 27) is due, as we have seen, to Theon. The two extra cases of XI. 23 were manifestly interpolated before Theon's time, for the preliminary distinction of three cases, ** (the centre) will either be within the triangle LMN^ or on one of the sides, or outside. First let it be within," is a spurious addition (B and V only). Similarly an unnecessary case is interpolated in III. II. H. Lemmas. Heiberg has unhesitatingly placed in his Appendix to Vol. III. certain lemmas interpolated either by Theon (on X. 13) or later writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas). But we are here concerned with the lemmas found in all the MSS., which however are, for different reasons, necessarily suspected. We will deal with the Book X. lemmas last (i) There is an a priori ground of objection to those lemmas which come after the propositions to which they relate and prove properties used in those propositions ; for, if genuine, they would be a sign of faulty arrangement such as would not be likely in a systematic work so carefully ordered as the Elements. The lemma to VI. 22 is one of this class, and there is the further objection to it that in VI. 28 Euclid makes an assumption which would equally require a lemma though none is found. The lemma after XII. 4 is open to the further objections that certain altitudes are used but are not drawn in the ^ An-Nairld, ed. Curtze, p. isi. 6o ^ INTRODUCTION [cH. v figure (which is not in the manner of Euclid), and that a peculiar expression ** parallele pi pedal solids described on (apaypatf^ofieva aw6) pristns** betrays a hand other than Euclid's. There is an objection on the score of language to the lemma after xni. 2. The lemmas on XL 23, XIII. 13, XIIL 18, besides coming after the propositions to which they relate, are not very necessary in themselves and, as regards the lemma to XIIL 13, it is to be noticed that the writer of a gloss in the proposition could not have had it, and the words "as will be proved afterwards** in the text are rightly suspected owing to differences between the MS, readings. The lemma to Xll* 2 also, to which Simson raised objection, comes after the proposition ; but, if it is rejected, the words ** as was proved before" used in XIL 5 and i8^ and referring to this lemma, must be struck out (2) Reasons of substance are fatal to the lemma before X 60, which is really assumed in X. 44 and therefore should have appeared there if anywhere, and to the lemma on X. 20, which tries to prove what is already stated in X. Det 4. We now come to the remaining lemmas in Book X., eleven in number, which come before the propositions to which they relate and remove difficulties in the way of their demonstration. That before X, 42 introduces a set of propositions with the words *' that the said irrational straight lines are uniquely divided >.,we will prove after premising the following lemma," and it is not possible to suppose that these words are due to an interpolator ; nor are there any objections to the lemmas before x, 14, 17^ 22, 33, 54, except perhaps that they are rather easy. The lemma before X. 10 and X io itself should probably be removed from the Elements ; for X, iO really uses the following proposition X. il, which is moreover numbered 10 by the firsthand in P, and the words in X, lo referring to the lemma "for we learnt (how to do this) '* betray the interpolator Heiberg gives reason also for rejecting the lemmas before X. 19 and 24 with the words "in any of the aforesaid ways " (omitted in the Theonine MSS.) in the enunciations of x. 19, 24 and in the exposition of X. 20. Lastly, the lemmas before x, 29 may be genuine, though there is an addition to the second of them which is spurious. Heiberg includes under this heading of interpolated lemmas two which purport to be substantive propositions, XL 38 and XIIL 6, These must be rejected as spurious for reasons which will be found in detail in my notes on XL 37 and xml 6 respectively. The latter proposition is only quoted once (in XIii. 17); probably the words quoting it (with 7/9a/i/Ai^ instead of eiOeia) are themselves interpolated, and Euclid thought the fact stated a sufficiently obvious inference from Xlll. I. III. Porisms (or corollaries). Most of the porisms in the text are both genuine and necessary ; but some are shown by differences in the MSS. not to be so, e.g. those to L 15 (though Proclus has it), III. 31 and VL 20 (For. 2). Sometimes parts of porisms are interpolated. Such are the last few lines in tlie porisms to iv. 5, yL 8 ; the latter addition is proved later by CH. v] THE TEXT 6i means of vi. 4, 8, so that the writer of these proofs could not have had the addition to vi. 8 For. before him. Lastly, interpolators have added a sort of proof to some porisms, as though they were not quite obvious enough ; but to add a demonstration is inconsistent with the idea of a porism, which, according to Proclus, is a by-product of a proposition appearing without our seeking it IV. Scholia. Several interpolated scholia betray themselves by their wording, e.g. those given by Heibei^ in the Appendix to Book X. and contain- ing the words KoKel^ iKoKeat ("he calls" or "called"); these scholia were apparently written as marginal notes before Theon's time, and, being I adopted as such by Theon, found their way into the text in P and I some of the Theonine MSS. The same thing no doubt accounts for I the interpolated analyses and syntheses to xiil. i — 5, as to which see \ my note on xill. i. f V. Interpolations in Book X. First comes the proposition " Let it be proposed to us to show that in square figures the diameter is incommensurable in length with the side," which, with a scholium after it, ends the tenth Book. The form of the enunciation is suspicious enough and the proposition, the proof of which is indicated by Aristotle and perhaps was Pythagorean, is perfectly unnecessary when X. 9 has preceded. The scholium ends with remarks about commensurable and incommensurable solids, which are of course out of place before the Books on solids. The scholiast on Book x. alludes to this particular scholium as being due to " Theon and some others." But it is doubtless much more ancient, and may, as Heiberg conjectures, have been the b^inning of Apollonius' more advanced treatise on incommensurables. Not only is everything in Book X. after X. 1 15 interpolated, but Heiberg doubts the genuineness even of x. 112 — 115, on the ground that x. in rounds ofT the theory of incommensurables as we want it in the Books on solid geometry, while x. 112 — 115 are not really connected with what precedes, nor wanted for the later Books, but seem to form the starting-point of a new and more elaborate theory of irrationals. VI. Other minor interpolations are found of the same character as those above attributed to Theon. First there are two places (XL 35 and XI. 26) where, after "similarly we shall prove " and "for the same reason," an actual proof is nevertheless given. Clearly the proofs are interpolated; and there are other similar interpolations. There are also interpolations of intermediate steps in proofs, unnecessary explanations and so on, as to which I need not enter into details. Lastly, following Heiberg's order, I come to VII. Interpolated definitions, axioms etc. Apart from VI. Def. 5 (which may have been interpolated by Theon although it is found written in the margin of P by the first hand), the definition of a segment of a circle in Book I. is interpolated, as is clear from the fact that it occurs in a more appropriate place in Book III. and Proclus omits it. \l. Def. 2 (reciprocal figures) is rightly condemned by Simson — perhaps it was taken from Heron — and i. i C9 INTRODUCTION [ch, v Heiber^ would reject VI I, Def lo, as to which see my note on that definition. Lastly the double definition of a solid angle (XL Def ii) constitutes a difficulty. The use of the word iTrt^idvtta suggests that the first definition may have been older than Euclid, and he may have quoted it from older iiemtnts^ especially as his own definition which follows only includes solid angles contained by planes, whereas the m, other includes other sorts (cf the words ^pap^fio^v, ypafi^<;) which are II also distinguished by Heron (Def. 24). If the first definition had i come last, it could have been rejected without hesitation : but it is not 1 so easy to reject the first part up to and including "otherwise" {aXXm<;). No difficulty need be felt about the definitions of '* oblong," "rhombus" "trapezium" and "rhomboid" which are not actually used in the Elematis; they were no doubt taken from earlier dements and given for the sake of completeness. As regards the axioms or, as they are called in the text, common notions {ttmvai ivvotai), it is to be observed that Proclus says* that Apollonius tried to prove ''the axioms/' and he gives Apollonius* attempt to prove Axiom k This shows at all events that Apollonius had some of the axioms now appearing in the text. But how could Apollonius have taken a controversial line against Euclid on the subject of axioms if these axioms had not been Euclid's to his know- ledge? Andi if they had been interpolated between Euclid's time and his own, how could Apollonius, living so comparatively short a time after Euclid, have been ignorant of the fact ? Therefore some of the axioms are Euclid's (whether he called them common notions, or axioms^ as is perhaps more likely since Proclus calls them axioms): and we need not hesitate to accept as genuine the first three discussed by Proclus, viz. (i) things equal to the same equal to one another, (2) if equals be added to equals, wholes equal, (5) if equals be subtracted from equals, remainders equal. The other two mentioned by Proclus (whole greater than part, and congruent figures equal) are more doubtful, since they are omitted by Heron, Martianus Capella, and others. The axiom that "two lines cannot enclose a space" is however clearly an interpolation due to the fact that I. 4 appeared to require it The others about equals added to unequals, doubles of the same thing, and halves of the same thing are also interpolated ; they are connected with other interpolations, and Proclus clearly used some source which did not contain them. Euclid evidently limited his formal axioms to those which seemed to him most essential and of the widest application ; for he not un- frequently assumes other things as axiomatic, e.g. in vil. 28 that, if a number measures two numbers, it measures their difference. The differences of reading appearing in Proclus suggest the Question of the comparative purity of the sources used by Proclus, leron and others, and of our text The omission of the definition of a s^ment in Book I. and of the old gloss ** which is called the cir- cumference" in L Def. 15 (also omitted by Heron, Taurus, Sextus * Pkoehu, pp. 194, loaqq. CH. v] THE TEXT 63 Empiricus and others) indicates that Proclus had better sources than we have ; and Heibci^ gives other cases where Proclus omits words which are in all our MSS. and where Proclus' reading should perhaps I be preferred. But, except in these instances (where Proclus may have drawn from some ancient source such as one of the older com- mentaries), Proclus' MS. does not seem to have been among the best Often it agrees with our worst MSS., sometimes it agrees with F where F alone has a certain reading in the text, so that (e.g. in I. 15 Por.) the common reading of Proclus and F must be rejected, thrice only does it agree with P alone, sometimes it agrees with P and some Theonine MSS., and once it agrees with the Theonine MSS. against P and other sources. Of the other external sources, those which are older than Theon generally agree with our best MSS., e.g. Heron, allowing for the difference in the plan of his definitions and the somewhat free adap- tation to his purpose of the Euclidean definitions in Books X., XI. Heiberg concludes that the Elements were most spoiled by inter- polations about the 3rd c, for Sextus Empiricus had a correct text, while lamblichus had an interpolated one; but doubtless the purer text continued for a long time in circulation, as we conclude from the fact that our MSS. are free from interpolations already found in lamblichus' MS. CHAPTER VL THE SCHOLIA- » ^ t , Heiberg has collected scholia, to the number of about 1500, in Vol V* of his edition of EucHd, and has also discussed and classified them in a separate short treatise, in which he added a few others*. These scholia cannot be regarded as doing much to facilitate the reading of the Eitntents. As a rule, they contain only such observa- tions as any intelligent reader could make for himself Among the few exceptions are XL Nos. 33, 35 (where XL 22^ 23 are extended to solid angles formed by any number of plane angles), XI L No. 85 (where an assumption tacitly made by Euclid in XIL 17 is proved), IX. Nos, 28, 29 (where the scholiast has pointed out the error in the text of JX. 19). Nor are they very rich in historical information ; they cannot be compared in this respect with Proclus' commentary on Book I* or with those of Eutocius on Archimedes and Apollonius, But even tinder this head they contain some things of interest, e.g. IL No. 1 1 explaining that the gnomon was invented by geometers for the sake of brevity, and that its name was suggested by an incidental characteristic^ namely that "from it the whole is known {jvmpi^^rai), ^Mh^t of the whole area or of the remainder, when it (the ywfL^v) is either placed round or taken away"; iL No. 13, also on the gnomon; IV, No. 2 stating that Book IV. was the discovery of the Pythagoreans ; V, Na I attributing the content of Book v, to Eudoxus; X. No. i with its allusion to the discovery of incommensurability by the Pytha- goreans and to Apollonius* work on irrationals ; X. No. 62 definitely attributing X, 9 to Theaetetus; XIIL No. i about the "Platonic" figures, which attributes the cube, the pyramid, and the dodecahedron to the Pythagoreans, and the octahedron and icosahedron to Theaetetus. Sometimes the scholia are useful in connexion with the settlement of the text, (i) directly^ e.g, IIL Na 16 on the interpolation of the word "within'* (iirri^) in the enunciation of IIL 6, and X, No. 1 alluding to the discussion by "Theon and some others" of irrational "surfaces" and '* solids," as well as '*lines/' from which we may ^ Heiberg, Om Stka/vrm tU EttkUth Elrminitr^ Kjifbenhft^Q, |S8S, Tbe tr^ct is wniten is Dankh, but, fortuniLt^ly for those who do not read Danish easilj, Lhe author has ftprpended (pp. 7^—78} a re&mne in French* CH.VI] THE SCHOLIA 65 conclude that the scholium at the end of Book x/ is not genuine ; (2) indirectly in that they sometimes throw light on the connexion of certain MSS. Lastly, they have their historical importance as enabling us to judge of the state of mathematical science at the times when they were written. Before passing to the classification of the scholia, Heiberg remarks that we must separate from them a number of additions in the nature of scholia which are found in the text of our MSS. but which can, in one way or another, be proved to be spurious. As they are found both in P and in the Theonine MSS., they must have been in the MSS. anterior to Theon (4th c). But they are, in great part, only found in the margin of P and the Theonine MSS.; in V they are half in the text and half in the margin. This can hardly be explained except on the supposition that these additions were originally (in the MSS. before Theon) in the margin, and that Theon kept them there in his edition, but that they afterwards found their way gradually into the text of P as well as of the Theonine MSS., or were omitted altogether, while particular MSS. have in certain places preserved the old arrange- ment Of such spurious additions Heiberg enumerates the following: the axiom about equals subtracted from unequals, the last lines of the porism to VI. 8, second porisms to V. 19 and to vi. 20, the porism to III. 31, VI. Def. 5, various additions in Book X., the analyses and syntheses of XIII. i — 5, and the proposition XIII. 6. The two first classes of scholia distinguished by Heiberg are denoted by the convenient abbreviations "SchoL Vat" and "Schol. Vind." L Schol. Vat. It is first necessary to set out the letters by which Heiberg denotes certain collections of scholia. P « Scholia in P written by the first hand. B a Scholia in B by a hand of the same date as the MS. itself, generally that of Arethas. F« Scholia in F by the first hand. Vat = Scholia of the Vatican MS. 204 of the loth c, which has these scholia on leaves 198 — 205 (the end is missing) as an independent collection. It does not contain the text of the Ekments, V*^ = Scholia found on leaves 283 — 292 of V and written in the same hand as that part of the MS. itself which begins at leaf 235. Vat 192 =s a Vatican MS. of the 14th c which contains, after (i) the Elements I. — XIli. (without scholia), (2) the Data with scholia, (3) Marinus on the Data^ the Schol. Vat as an independent collection and in their entirety, b^inning with I. No. 88 and ending with Xlll. No. 44. The Schol. Vat, the most ancient and important collection of scholia, comprise those which are found in PBF Vat. and, from vil. 12 . to IX. 1 5, in PB Vat. only, since in that portion of the Elements F was restored by a later hand without scholia ; they abo include I. H. £. e . 66 INTRODUCTION [ch. vi No. 88 which onfy happens to be erased in F, and IX. Nos, 28, 29 which may be left out because F here has a different text In F and Vat the collection ends with Book X, ; but it must also include SchoL PB of Books XL^ — xilt, since these are found along with Schol* Vat to Books L— X* in several MSS. (of which Vat 192 is one) as a separate collection. The SchoL Vat to Books X,— XIIL are also found in the collection V^ (where, curiously enough, Xllt Nos. 45, 44 are at the beginning). The Scho!, Vat. accordingly include SchoL PBV*= Vat 193, and doubtless also those which are found in two of these sources. The total number of scholia classified by Heiberg as SchoL Vat is 138, As r^ards the contents of SchoL Vat Heiberg has the following observations. The thirteen scholia to Book T. are extracts made from Proclus by a writer thoroughly conversant with the subject, and cleverly recast (with some additions). Their author does not seem to have had the two lacunae which our text of Proclus has (at the end of the note on t 36 and the beginning of the next note^ and at the beginning of the note on L 43), for the scholia L Nos. 125 and 137 seem to hll the gaps appropriately, at least in part. In some passages he had better readings than our MSS. have. The rest of SchoL Vat (on Books ll. — XIII.) are essentially of the same character as those on Book I., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas; they are, in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, Heiberg places some of the SchoL Vat to Book x, side by side with passages from the com- mentary of Pappus in the Arabic translation discovered by Woepcke* ; he also refers to the striking confirmation afiforded by the fact that XIL No. 2 contains the solution of the problem of inscribing in a given circle a polygon similar to a polygon inscribed in another circle, which problem Eutocius says' that Pappus gave in his commentary on the Ekmettts. But, on the other hand, SchoL Vat. contain some things which cannot have come from Pappus, e.g. the allusion in X. No, i to Theon and irrational surfaces and solids, Theon being later than Pappus ; lit No, 10 about porisms is more like Proclus' treatment of the subject than Pappus', though one expression recalls that of Pappus about/ormtn^ (axnM^ri^eaOai) the enunciations of porisms like those of either theorems or problems. The Schol. Vat give us important indications as regards the text of the Elements as Pappus had it In particular, they show that he could not have had in his text certain of the lemmas in Book x. For example, three of these are identical with what we find in Schol. ^ Om ScMitrm tU Euklids Elenunttr^ pp. 11, 13 : cf. EukHd'Stutiien^ pp. 170, 171 ; Woepcke, Mimciru friunt. d tAcad, da Scufues, 1856, xiv. p. 658 sqq. * Archimedes, ed. Hdbeig, iiL p. 54, 5 — 8. I J (| ; I CH. VI] THE SCHOLIA 67 Vat (the lemma to x. i7 = Schol. x. No. 106, and the lemmas to • X. 54, 60 come in Schol. X. No. 328) ; and it is not possible to suppose * \ that these lemmas, if they were already in the text, would also be given as scholia. Of these three lemmas, that before X. 60 has already been condemned for other reasons; the other two, un- objectionable in themselves, must be rejected on the ground now stated. There were four others against which Heiberg found nothing to urge when writing his prolegomena to Vol. v., viz. the lemmas before X. 42, x. 14, x. 22 and X. 33. Of these, the lemma to X. 22 is not reconcilable with Schol. X. No. 161, which takes up the assumption in the text of Eucl. x. 22 as if no lemma had gone before. The lemma to X. 42, which, on account of the words introducing it (see p. 60 above), Heiberg at first hesitated to regard as an inter- polation, is identical with Schol. X. No. 270. It is true that in Schol. X. No. 269 we find the words "this lemma has been proved before (iv roh ifiirpoaOev), but it shall also be proved now for convenience' sake (rov iroifiov tv€KaY and it is possible to suppose that " before " may mean in Euclid's text before X. 42 ; but a proof in that place would surely have been as ** convenient " as could be desired, and it is therefore more probable that the proof had been given by Pappus in some earlier place. (It may be added that the lemma to x. 14, which is identical with the lemma to XI. 23, con- demned on other grounds, is for that reason open to suspicion.) Heiberg's conclusion is that all the lemmas are spurious, and that most or all of them have found their way into the text from Pappus' commentary, though at a time anterior to Theon's edition, since they are found in all our MSS. This enables us to fix a date for these interpolations, namely the first half of the 4th c. Of course Pappus had not in his text the interpolations which, from the fact of their appearing only in some of our MSS., are seen to be later than those above-mentioned. Such are the lemmas which are found in the text of V only after x. 29 and x. 31 respectively and are given in Heiberg's Appendix to Book x. (numbered 10 and 11). On the other hand it appears from Woepcke's tract^ that Pappus already had X. 115 in his text : though it does not follow from this that the proposition is genuine but only that interpolations began very early. Theon interpolated a proposition (or lemma) between X. 12 and X. 13 (No. 5 in Heiberg*s Appendix). Schol. Vat. has the same thing (X. No. 125). The writer of the scholia therefore did not find this lemma in the text. Schol. Vat. IX. Nos. 28, 29 show that neither did he find in his text the alterations which Theon made in Eucl. IX. 19; the scholia in fact only agree with the text of P, not with Theon's. This suggests that Schol. Vat. were written for use with a MS. of the ante-Theonine recension such as P is. This probability is further confirmed by a certain independence which P shows in several places when compared with the Theonine MSS. Not only has P better readings in some passages, but more substantial divergences, and, * Woepcke, op, cit. p. 70a. 5—2 ».i. » introi>ucti6n in particularp the absence in P of three notes of a historical character which are added ^ wholly or partly from Proclus, in the Theonine MSS, attests an independent and more primitive point of view in P, In view of the distinctive character of P, it is possible that some of the scholia found in it in the first hand, but not in the other sources of SchoK Vat, also belong to that collection ; and several circumstances confirm this. Schol, XI I L No. 45. found in P only, which relates to a passage in Eucl Xill. 15, shows that certain words in the text, though older than Theon» are interpolated ; and, as the scholium is itself older than Theon, is headed *' tkird lemma/' and follows a "second lemma" relating to a passage in the text im- mediately preceding, which ** second lemma" belongs to Schol Vat and is taken from Pappus, the *'third" in all probability came from Pappus also. The same is true of SchoL XlL No. 72 and XII L No* 69, which are respectively identical with the propositions t^u/^o XL 38 (Heiberg, App, to Book XL, No. 5) and xlll. 6; for both of these interpolations are older than Theon, Moreover most of the scholia which P in the first hand alone has are of the same character as Schoh Vat Thus VIL No. 7 and XIIL No. i introducing Books vil. and XIIL respectively are of the same historical character as several of SchoL Vat, ; that Vll, No. 7 appears in the t€zt of P at the beginning of Book vil constitutes no difficulty. There are a number of converses^ remarks on the relation of propositions to one another, explanations such as XIL No. 89 in which it is remarked that 4>, H in Euclid's figure to XIL 17 {Z^ V in my figure) are really the same point but that this makes no difference in the proof. Two other Schol. P on XIL 17 are connected by their headings with xn. No. 72 mentioned above, XL No, 10 (P) is only another form of XL No, 1 1 (B) ; and B often, alone with P, has preserved Schol. Vat On the whole Heiberg considers some 40 scholia found in P alone to belong to Schol. Vat The history of Schol, Vat. appears to have been, in its main outlines, the following. They were put together after 500 A<D., since they contain extracts from Proclus, to which we ought not to assign a date too near to that of Proclus' work itself; and they must at least be earlier than the latter half of the 9th c, in which B was written. As there must evidently have been several intermediate links between the archetype and B, we must assign them rather to the first half of the period between the two dates, and it is not improbable that they were a new product of the great development of mathematical studies at the end of the 6th c. (Isidorus of Miletus). The author extracted what he found of interest in the commentary of Proclus on Book L and in that of Pappus on the rest of the work, and put these extracts in the margin of a MS, of the class of P, As there are no scholia to L I — 22, the first leaves of the archetype or of one of the eariiest copies must have been lost at an early date, and it was from that mutilated copy that partly P and partly a MS. of the Theonine class were taken, the scholia being put in the margin in both. Then the collection spread through the Theonine MSS., gradually losing some CH.VI] THE SCHOLIA 69 I ca.Y I scholia which could not be read or understood, or which were I accidentally or deliberately omitted. Next it was extracted from \ one of these MSS. and made into a separate work which has been I preserved, in part, in its entirety (Vat 192 etc.) and, in part, divided I into sections, so that the scholia to Books X. — XIII. were detached ; (V^). It had the same fate in the MSS. which kept the original arrangement (in the mai^in), and in consequence there are some MSS. where the scholia to the stereometric Books are missing, those Books having come to be less read in the period of decadence. It is from one of these MSS. that the collection was extracted as a separate work such as we find it in Vat. (loth c). II. The second great division of the scholia is SchoL Vind. This title is taken from the Viennese MS. (V), and the letters used by Heiberg to indicate the sources here in question are as follows. V* = scholia in V written by the same hand that copied the MS. itself from fol. 235 onward. q=s scholia of the Paris MS. 2344 (q) written by the first hand. 1 = scholia of the Florence MS. Laurent XXVlii, 2 written in the 13th-- 14th c, mostly in the first hand, but partly in two later hands. V^s= scholia in V written by the same hand as the first part (leaves i— 183) of the MS. itself; V^ wrote his scholia after V*. q* = scholia of the Paris MS. (q) found here and there in another hand of early date. Schol. Vind. include scholia found in VH). 1 is nearly related to q ; and in fact the three MSS. which, so far as Euclid's text is con- cerned, show no direct interdependence, are, as regards their scholia, derived from one original. Heiberg proves this by reference to the readings of the three in two passages (found in Schol. I. No. 109 and X. No. 39 respectively). The common source must have contained, besides die scholia found in the three MSS. V*ql, those also which are contained in two of them, for it is more unlikely that two of the three should contain common interpolations than that a particular scholium should drop out of one of them. Besides V* and q, the scholia V^ and q' must equally be referred to Schol. Vind., since the greater part of their scholia are found in 1. There is a lacuna in q from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements Schol. Vind. are represented by VI only. Heiberg pves about 450 numbers in all as belonging to this collection. Schol. Vind. did not all come from one source ; this is shown by diflferences of substance, e.g. between x. Nos. 36 and 39, and by differences of time of writing : e.g. VI. No. 52 refers at the b^inning to No. 55 with the words "as the scholium has it" and is therefore later than that scholium ; X. No. 247 is also later than X. No. 246. The scholia to Book I. are here also extracts from Proclus, but more copious and more verbatim than in Schol. Vat The author has not always understood Proclus; and he had a text as bad as that of our MSS., with the same lacunae. The scholia to the other INTRODUCTION ^> [ch. vi Mil*' *■ 3^f*; Books are partly drawn (r) from Schol Vat, the MSS. representing Schol Vind. and Schol. Vat in these cases showing nearly all possible combinations; but there is no certain trace in Schol Vind< of the scholia peculiar to P. The author used a copy of Schol Vat, in the form in which they were attached to the Theonine text ; thus Schol Vind. correspond to BF Vat, where these diverge from P, and especially closely to B, Besides Schol Vat., the editors of Schol Vind. used (2) other old collections of scholia of which we find traces in B and F; Schol Vind, have also some scholia common with b. The scholia which Schol Vind, have in common with BF come from two different sources, and were apparently afterwards introduced into the other MSS. ; one result of this is that several scholia are reproduced twice. Butj besides the scholia derived from these sources, Schol Vind contain a large number of others of late date, characterised by in- correct language or by triviality of content (there are many examples in numbers, citations of propositions used, absurd diropiai, and the hke). Unlike Schol Vat, these scholia often quote words from Euclid as a heading (in one case a heading is inserted in Schol Vind. where a scholium without the heading is quoted from Schol Vat, see V. No. 14). The explanations given often presuppose very little know- ledge on the part of the reader and frequently contain obscurities and gross errors. Schol Vind* were collected for use with a US. of the Theonine class; this follows from the fact that they contain a note on the proposition vuigo VII. 22 interpolated by Theon (given in Heiberg's A pp. to Vol II. p. 430). Since the scholium to vil 39 given in V and p in the text after the title of Book VIIL quotes the proposition as VII. 39, it follows that this scholium must have been written before the interpolation of the two propositions vu^a VII. 20, 22 ; Schol Vind. contain (viL No* 80) the first sentence of it, but without the heading referring to VI l* 39* Schol VIL No* 97 quotes vlL 33 as vn* 34, so that the proposition vui^ VII. 22 may have stood in the scholiast's text but not the later interpolation vu/go VIL 20 (later because only found in B in the margin by the first hand). Of course the scholiast had also the interpolations earlier than Theon* For the date of the collection we have a lower limit in the date (12th c.) of MSS. in which the scholia appear* That it was not much earlier than the 1 2th c. is indicated ( r ) by the poverty of its contents, (2) by the quality of the MS. of Proclus which was used in the compilation of it (the Munich MS. used by Friedlein with which the scholiast's excerpts are essentially in agreement belongs to the i ith — t2th c.)^ (3) by the fact that Schol Vind. appear only in uss. of the 1 2th c* and no trace of them is found in our MSS, belonging to the 9th — loth c* in which Schol Vat are found. The collection may therefore probably be assigned to the iith c. Perhaps it may be in part due to Psellus who lived towards the end of that century : for in a Florence MS. (Magltabecch. Xl^ 53 of the 15th c) containing a mathematical compendium intended for use in the reading of Aristotle CH. VI] THE SCHOLIA 71 the scholia I. Nos. 40 and 49 appear with the name of Psellus attached. f.* Schol. Vind. are not found without the admixture of foreign M elements in any of our three sources. In 1 there are only very few such in the first hand. In q there are several new scholia in the first hand, for the most part due to the copyist himself The collection of scholia on Book X. in q (Heiberg's q^) is also in the first hand ; it is not original, and it may perhaps be due to Psellus (Maglb. has some definitions of Book X. with a heading "scholia of... Michael Psellus on the definitions of Euclid's loth Element'' and Schol. x. No. 9), whose name must have been attached to it in the common source of Maglb. and q ; to a great extent it consists of extracts from Schol. Vind. taken from the same source as VI. The scholia q* (in an ancient hand in q), confined to Book II., partly belong to Schol. Vind. and partly correspond to W (Bologna MS.), q* and q^ are in one hand (Theodorus Antiochita), the nearest to the first hand of q ; they are doubtless due to an early possessor of the MS. of whom we know nothing more. V* has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat 1038, 13th c.) ; these scholia wei-e taken from a source in which many abbreviations were used, as they were often misunderstood by V". Other scholia in V* which are hot found in the older sources — some appearing in V* alone — are also not original, as is proved by mistakes or corruptions which they contain ; some others may be due to the copyist himself. V^ seldom has scholia common with the other older sources ; for the most part they either appear in V* alone or only in the later sources as v or P (later scholia in F), some being original, others not In Book x. V^ has three series of numerical examples, (i) with Greek numerals, (2) alternatives added later, also mostly with Greek numerals, (3) with Arabic numerals. The last class were probably the work of the copyist himself. As V^ belongs to the same time as the MS. (i2th c), these examples give an idea of the facility with which calculations were made in Byzantium at that time. They show too that the Greek method of writing numbers still preponderated in the nth c, but that the use of the Arabic numerals (in the East- Arabian form) was thoroughly established in the 12th c Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V* has very few scholia which are found in other sources, the greater part being original ; V*, V* are the work of the copyist himself; V^ are so in part only, and contain several scholia from Schol. Vat. and other sources. V* and V* are later than 13th — 14th c, since they are not found in f (cod. Laurent. XXVIII, 6) which was copied from V and contains, besides V* V^ the greater part of V^ and VI. No. 20 of V* (in the text). In P there are, besides P* (a quite late hand, probably one of the old Scriptores Graeci at the Vatican), two late hands (P*), one of which has some new and independent scholia, while the other has 72 INTRODUCTION added the greater part of Schol Vm±, partly in the margin and partly on pieces of leaves stitched on, Our sources for Schol Vat also contain other elements. In P there were introduced a certain number of extracts from Proclus, to supplement Schol. Vat, to Book l. ; they are all written with a different ink from that used for the oldest part of the MS,, and the text is inferior. There are additions in the other sources of Schol Vat (F and B) which point to a common source for FB and which are nearly all found in other MSa, and, in particular, in Schol Vind., which also used the same source ; that they are not assignable to Schol Vat results only from their not being found in Vat Of other additions in F, some are peculiar to F and some common to it and b; but they are not original F* (scholia in a later hand in F) contains three original scholia ; the rest come from V. B contains, besides scholia common to it and F, b or other sources, several scholia which seem to have been put together by Arethas, who wrote at least a part of them with his own hand. Heiberg has satisfied himself, by a closer study of b, that the scholia which he denotes by b, fi and b* are by one hand ; they are mostly to be found in other sources as well though some are original By the same hand (Theodorus Cabasilas, 1 5th c.) are also the scholia denoted by W, B*, b* and B*. These scholia come in great part from Schol Vind., and in making these extracts Theodorus probably used one of our sources, 1, mistakes in which often correspond to those of Theodorus, To one scholium is attached the name of Demetrius (who must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c*); but it could not have been written by him, since it appears in B and Schol Vind. Nor are all the scholia which bear the name of Theodorus due to Theodorus himself, though some are so. As B* (a late hand in B) contains several of the original scholia of b", B* must have used b itself as his source, and, as all the scholia in B* are in b, the latter is also the source of the scholia in B* which are found in other MSS. B and b were therefore, in the 1 5th c, in the hands of the same person ; this explains, too, the fact that b in a late hand has some scholia which can only come from B, We arrive then at the conclusion that Theodorus Cabasilas, in the 15th c, owned both the MSS. B and b, and that he transferred to B scholia which he had before written in b, either independently or after other sources, and inversely transferred some scholia from B to b. Further, B' are earlier than Theodorus Cabasilas, who certainly himself wrote B* as well as b* and b*. An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c) along with scholia on I. 31, x. 14 and x. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heiberg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used 1 as his copy. There are also in 1 two other byzantine scholia, written by a late hand, and bearing the names loannes and Pediasimus respectively ; r t CH.VI] THE SCHOLIA 73 these must in like manner have been written by a pupil after lectures of loannes Pediasimus (first half of 14th c), and this pupil must also have used 1. Before these scholia were edited by Heiberg, very few of them had been published in the original Greek. The Basel editio princeps has a few (v. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken, some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others probably from the Venice MS. (Marc. 301) also used by him; one published by Heiberg, not in his edition of Euclid but in his paper on the scholia, may also be from Venet. 301, but appears also in Paris. Gr. 2342. The scholia in the Basel edition passed into the Oxford edition in the text, and were also given by August in the * Appendix to his Vol. II. I Several specimens of the two series of scholia (Vat. and Vind.) ' were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) I and by KnocYit {Untersuchungen fiber die neu aufgefundenen Scholien des Proklus, Herford, 1865). The scholia published in Latin were much more numerous. G. Valla {De expetettdis et fugiendis rebus^ 1501) reproduced apparently some 200 of the scholia included in Heiberg's edition. Several of these he obtained from two Modena MSS. which at one time were in his possession (Mutin. Ill B, 4 and II E, 9, both of the 15th c); but he must have used another source as well, containing extracts from other series of scholia, notably Schol. Vind. with which he has some 87 scholia in common. He has also several that are new. Commandinus included in his translation under the title *' Scholia antiqua" the greater part of the Schol. Vat. which he certainly obtained from a MS. of the class of Vat. 192; on the whole he adhered closely to the Greek text Besides these scholia Com- mandinus has the scholia and lemmas which he found in the Basel editio princeps^ and also three other scholia not belonging to Schol. Vat, as well as one new scholium (to xil. 13) not included in Heiberg's edition, which are distinguished by different type and were doubtless taken from the Greek MS. used by him along with the Basel edition. In Conrad Dasypodius' Lexicon mathematicum published in 1573 there is (on fol. 42 — 44) "Graecum scholion in definitiones Euclidis libri quinti elementorum appendicis loco propter pagellas vacantes annexum." This contains four scholia, and part of two others, published in Heiberg's edition, with some variations of readings, and with some new matter added (for which see pp. 64 — 6 of Heiberg's pamphlet). The source of these scholia is revealed to us by another work of Dasypodius, Isaaci Monachi Scholia in Euclidis elententorum geometriae sex priores libros per C Dasypodium in latinutn sermonem translata et in lucent edita (1579). This work contains, besides excerpts from Proclus on Book I. (in part closely related to Schol. Vind.), some 30 scholia included in Heiberg's edition, several new scholia, and the above-mentioned scholia to the definitions of Book V. published in Greek in 1573. After the scholia follow ** Isaaci Monachi 74 INTRODUCTION [c», vr prolegomena in Eudidis Elementorum geometriae Hbros" (two definitions of geometry) and "Van a miscellanea ad geometriae cogni- tionem necessana ab Isaaco Monacho collecta" (mostly the same as pp, 252^ 24 — 272, 27 in the Variae Colkcticnes included in Hultsch's Heron); lastly, a note of Dasypodius to the reader says that these schoha were taken '^ex clarissimi vrri Joan n is Sambuci anttquo cod ice many propria Isaaci Monachj scripto/* Isaak Monachus is doubtless Isaak Argyrus, 14th c; and Dasypodtus used a MS. in which, besides the passage in Hultsch's Variae CQikctwfus, were a number of scholia marked in the margin with the name of Isaak (c£ those in b under the name of Theodorus Cabasilas). Whether the new schoJia are original cannot be decided until they are published in Greek ; but it is not improbable that they are at all events independent arrange- ments of older scholia. All but five of the others, and all but one of the Greek scholia to Book v.^ are taken from Schol. Vat ; three of the excepted ones are from Schol, Vind,, and the other three seem to come from F (where some words of them are illegiblej but can be supplied by means of Mut in B, 4, which has these three scholia and generally shows a certain likeness to Isaak s scholia). Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c) on Eucl Book ll., which finds a place in Appendix IV, to the Scholia in Heiberg s edition. - CHAPTER VII. EUCLID IN ARABIA. We are told by Hajl Khalfa* that the Caliph al-Mansur (754-775) I sent a mission to the Byzantine Emperor as the result of which he (obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma'mun (813-833) obtained manuscripts of * Euclid, among others, from the Byzantines. The version of the } Elements by al-Hajjaj b. Yusuf b. Matar is, if not the very first, at least one of the first books translated from the Greek into Arabic*. According to the Fihrist^ it was translated by al-Hajjaj twice; the first translation was known as " Haruni " (" for Harun "), the second I bore the name "Ma'muni" ("for al-Ma'mun") and was the more trust- worthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, i) which is being published by Besthom and Heiberg*. In the preface to this MS. it is stated that, in the reign of Harun ar-RashId (786-809), al-Hajjaj was commanded by Yahya ! b. Khalid b. Barmak to translate the book into Arabic Then, when ^ al-Ma'mun became Caliph, as he was devoted to learning, al-Hajjaj saw that he would secure the favour of al-Ma'mun "if he illustrated and '. expounded this book and reduced it to smaller dimensions. He I' accordingly left out the superfluities, filled up the gaps, corrected or ' removed the errors, until he had gone through the bcK)k and reduced it, when corrected and explained, to smaller dimensions, as in this I copy, but without altering the substance, for the use of men endowed • with ability and devoted to learning, the earlier edition, being left in '• the hands of readers." The FUifist goes on to say that the work was next translated by Ishaq b. Hunain, and that this translation was improved by Thabit b. Qurra. This Abu Ya'qub Ishaq b. Hunain b. Ishaq al-'ibadi (d. 910) was the son of the most famous of Arabic translators, Hunain b. Ishaq al-'ibadi (809-873), a Christian and physician to the Caliph al- Mutawakkil (847-861). There seems to be no doubt that Ishaq, who ^ Lexicon bibliogr. et encychp, ed. FlUgel, III. pp. 91, 93. • Klamroth, Zeitschrift dtr Deutschen Morgenldndischen GeseUschafU XXXV. p. 303. ' » Fihrist (tr. Sutcr), p. 16. ^ Codtx Leidensis 399, i . Euclidis EUmenta ex interfretaiume al'Hadschdschadschii cum • ccmmeniariis al'NarizU^ Hauniae, part I. i. 1893, part I. ii. 1897, part II. i. 1905. i' r6 INTRODUCTION [ra. vu must have known Greek as well as hts father, made his translation direct from the Greek, The revision must apparently have been the subject of an arrangement between Ishaq and T ha bit, as the latter died in 90 ! or nine years before Ishaq, Thabit undoubtedly consulted Greek MSS. for the purposes of his revision. This is expressly stated in a marginal note to a Hebrew version of the Elements^ made from Ishaq's, attributed to one of two scholars belonging to the same family ^ viz, either to Moses b. Tibbon (about 1244-1274) or to Jakob b, Machir (who died soon after 1306)^ Moreover Thabit observes, on the pro- position which he gives as ix. 3 1, that he had not found this proposition and the one before it in the Greek but only in the Arabic ; from which statement Klamroth draws two conclusions, ( 1 ) that the Arabs had already begun to interest themselves in the authenticity of the text and (2) that Thabit did not alter the numbers of the propositions in Ishaq's translation'. The Fihrisi also says that Yuhanna al-Qass (i.e. " the Priest ") had seen in the Greek copy in his possession the pro- position in Book I. which Thabit took credit for, and that this was confirmed by Nazlf, the physician, to whom Yuhanna had shown it This proposition may have been wanting in Ishaq, and Thabit may have added it, but without claiming it as his own discovery'. As a fact, I. 45 is missing in the translation by al-Hajjaj, The original version of Ishaq wtthcut the improvements by ThSbit has probably not survived any more than the first of the two versions by al'Hajjaj ; the divergences between the MSS. are apparently due to the voluntary or involuntary changes of copyists, the former class varying according to the degree of mathematical knowledge possessed by the copyists and the extent to which they were influenced by considerations of practical utility for teaching purposes*. Two MSS* ' of the Ishaq -Thabit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in i26o~i)"; Books L — XIIL are in the Ishaq-Thabit version, the non-Euclidean Books XIV,, XV. in the translation of Qusta b. Luqa al-Ba'labakkl (d. about 912)* The first of these MSS. (No- 279) is that (O) used by Klamroth for the purpose of his paper on the Arabian Euclid. The other MS. used by Klamroth is (K) Kj^enhavn LXXXI, undated but probably of the 13th c, containing Books V. — XV., Books v. — X. being in the Ishaq-Thabit version, Books XL — XIII. purporting to be in al-Hajjaj's translation, and Books xiv., XV. in the version of Qusta b. Luqa. In not a few propositions K and O show not the slightest difference, and, even where the proofs show considerable diflferences, they are generally such that, by a careful comparison, it is possible to reconstruct the common archetype, so that it is fairly clear that we have in these cases, not two recensions of one translation, but arbitrarily altered and ^ Steinfchneider, ZeiUchrift fiir Math, u, Pkysik, xxxi., hist-litt Abtheilung, pp. 85, "Klamroth, p. 379. * Stdmchndder, p. 88. ^ Klamroth, p. 300. * These mss. are described by NicoU and Pusey, Caiaiogus cod, mss, orimi. diN. Bod- Umttoi, pt IL 1835 (pp. 357 — 262). I CH. vii] EUCLID IN ARABIA 77 shortened copies of one and the same recension ^ <i^The Bodleian MS. No. 280 contains a preface, translated by Nicoll, which cannot be by Thabit himself because it mentions Avicenna (980-1037) and other later authors. The MS. was written at Mara^a in the year 1 260-1 and . has in the margin readings and emendations from the edition of f Naslraddin at-TusI (shortly to be mentioned) who was living at Maraga ( at the time. Is it possible that at-TusI himself is the author of the I preface'? Be this as it may, the preface is interesting because it throws light on the liberties which the Arabians allowed themselves ^ to take with the text. After the observation that the book (in spite r of the labours of many editors) is not free from errors, obscurities, L redundancies, omissions etc, and is without certain definitions neces- I sary for the proofs, it goes on to say that the man has not yet been ' found who could make it perfect, and next proceeds to explain (i) that Avicenna *'cut out postulates and many definitions" and r attempted to clear up difficult and obscure passages, (2) that Abu'l I Wafa al-Buzjanl (939-997) "introduced unnecessary additions and I left out many things of great importance and entirely necessary," inasmuch as he was too long in various places in Book VI. and too short in Book X. where he left out entirely the proofs of the a^tamae, while he made an unsuccessful attempt to emend XII. 14, (3) that Abu Ja'far al-Khazin (d. between 961 and 971) arranged the postulates excellently but " disturbed the number and order of the propositions, reduced several propositions to one " etc. Next the preface describes the editor's own claims* and then ends with the sentences, " But we have kept to the order of the books and propositions in the woric itself (i.e. Euclid's) except in the twelfth and thirteenth books. For we have dealt in Book xiil. with the (solid) bodies and in Book Xll. with the surfaces by themselves." After Thabit the Fihrist mentions Abu 'Uthman ad-Dimashq! as having translated some Books of the Elements including Book X. (It is Abu 'Uthman's translation of Pappus* commentary on Book x. which Woepcke discovered at Paris.) The Fihrist adds also that " Nazlf the physician told me that he had seen the tenth Book of Euclid in Greek, that it had 40 propositions more than the version in common circulation which had 109 propositions, and that he had determined to translate it into Arabic." But the third form of the Arabian Euclid actually accessible to us is the edition of Abu Ja'far Muh. b. Muh. b. al-Hasan Naaraddin at-TusI (whom we shall call at-TusI for short), bom at Tus (in Khurasan) in 1201 (d. 1274). This edition appeared in two forms, a larger and a smaller. The larger is said to survive in Florence only (Pal. 272 and 313, the latter MS. containing only six Books) ; this was published at Rome in 1 594, and, remarkably enough, some copies of ' Klamroth, pp. 306—8. * Steinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitichrift . , fiir Math, u. Physik, XXIX., hist.-Iitt. AUh. p. 16. * This seems to include a rearrangement of the contents of Books xiv., xv. added to the t Elements, > • V- ^ ...^ * Cnrtse, 0^. ^. p. so; Heibeig, EukUd-Studim^ p. 178. • Hdberg^s Eudidy voL v. p. ci. *• Klamrotb, pp. 173— 4. ft INTRODUCTIOli^ [cb. ¥ii , this edition are to be found with 12 and some with 13 Books, some with a Latin title and some without\ But the book was printed tn Arabic, so that K^stner remarks that he will say as much about it as can be said about a book which one cannot read* The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334*, 1335X P^ns (2465. 2466), India Office, and Constantinople ; it was printed at Constantinople in 1801, and the first six Books at Calcutta in i824\ At*Tusl's work is however not a translation of Euclid's text, but a re*wr!tten Euclid based on the older Arabic translations. In this respect it seems to be like the Latin version of the EUmt^nts by Campanus {Cam pan o), which was first published by Erhard Ratdolt at Venice in J 482 (the first printed edition of Euclid*). Campanus (13th c.) was a mathematician^ and it is likely enough that he allowed himself the same liberty as at-TusI in reproducing Euclid. What- ever may be the relation between Campanus' version and that of Athelhard of Bath (about 1 120), and whether, as Curt^e thinks'^ they both used one and the same Latin version of loth — 1 Ith c, or whether Campanus used Athclhard's version in the same way as at-Tusi used those of his predecessors', it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in them I Campanus' version is not of much service for the purpose of forming a judgment on the relative authenticity of the Greek and Arabian tradition ; but it sometimes preserves traces of the purer source, as when it omits Theons addition to wu 33'* A curious circumstance is that, while Campanus' version agrees with at-Tust's in the number of the propositions in all the genuine Euclidean Bcwks except V. and IX., it agrees with Athelhard's in having 34 propositions i in Book V. (as against 25 in other versions), which confirms the view ; that the two are not independent, and also leads, as Klamroth says, to this dilemma: either the additions to Book v. are Athelhard's own, or he used an Arabian Euclid which is not known to us^. Heiberg also notes that Campanus' Books xiv., xv. show a certain agreement with the preface to the Thabit-Ishaq version, in which the author claims to have (1) given a method of inscribing spheres in the five r^ular solids, (2) carried further the solution of the problem how ' Sater, Die Mathematikir und Astronomm der Arader, p. 151. The Latin title is Eiulidis iUmeniorum geomeirkorum Wnri tredeeim. Ex traJitione doctissimi Nasiridim I Tktmi nunc frimnm arabice impressu Romae in typographia Medicea MDXCiv. Cum lioentia sapenonim. { * KMstner, Geschukie der Maihemaiik^ I. p. 367. I * Smer has a note that this MS. is very old, having been copied from the original in the y anthor's lifetime. ' * Suter, p. 151. * Described by Kistner, Geschickte der Mathemaiik^ i. pp. S89--399, and by Weiss- enbom, Die Ubersettungm da EukHd durch Campano und Zamberti^ Halle a. S., 188s, pp. I— -7. See also in/rat Chapter vni, p. 97. * Sonderabdruck des Jakresberichies iiSer die Fortschritte der kiassiuhen AUerthuwn- wissenschfai vom Oki. 1879—1881, Berlin, 1884. ' Klamroth, p. 371. CH. vii] EUCLID IN ARABIA 79 to inscribe any one of the solids in any other and (3) noted the cases where this could not be done\ With a view to arriving at what may be called a common measure of the Arabian tradition, it is necessary to compare, in the first place, the numbers of propositions in the various Books. Hajl Khalfa says that al-Hajjaj's translation contained 468 propositions, and Thabit's 478 ; this is stated on the authority o( at-Tusi, whose own edition contained 468*. The fact that Thabit's version had 478 propositions is confirmed by an index in the Bodleian MS. 279 (called O by Klamroth). A register at the beginning of the Codex Leidensis 399, i which gives Ishaq's numbers (although the translation is that of al-Hajjaj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 1 1, instead of the 10 of O*). I subjoin a 1 table ot relative nu the corresponding mbers tal numbers <en trom Kl in August's amroth, to which I have ad and Heiberg*s editions of 1 Greek text. - 1 The Arabian Euclid The Greek Euclid Books Ishaq at-TusI Campanus Gregory August Heiberg 1 I 48 48 48 48 48 48 II 14 14 14 14 14 14 • III 36 36 36 37 37 37 IV 16 16 16 16 16 16 V 25 25 34 25 25 25 VI 33 32 32 33 33 33 VII 39 39 39 41 41 39 VIII 27 25 25 27 . ^7 27 IX 38 36 39 36 36 36 X 109 107 107 117 116 "5 XI 41 41 41 40 40 39 XII 15 IS 15 18 18 18 XIII 21 18 18 18 18 18 462 452 464 470 469 465 [xiv 10 10 18 7 ? XV 6 6 13 10 478 468 495 487 ?] The numbers in the case of Heiberg include all propositions which he has printed in the text; they include therefore XIII. 6 and III. 12 now to be regarded as spurious, and X. 112 — 115 which he brackets as doubtful. He does not number the propositions in Books XIV., xv., but I conclude that the numbers in P reach at least 9 in XIV., and 9 in XV. > Heiberg, Zeitichrift fur Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. i\, • Klamroth, p. 274; Steinschneider, Zeitichrift fiir Math, u, Physti, xxxi., hist.-litt. Abth. p. 98. * Besthom- Heiberg read ** 11 ?" asihe number, Klamroth had read it as 31 (p. 373). 1 80 INTRODUCTION [ch. to The Fihrist confirms the number 109 for Book X,, from which Klamroth concludes that Ishaq's version was considered as by far the most authoritative. In the text of O, Book iv, consists of 17 propositions and Book XIV, of 1 2, differing in this respect from its own table of contents ; IV. 15, 16 in O are really two proofs of the same proposition. In al-Hajjaj's version Book L consists of 47 propositions only, 1, 45 being omitted. It has also one proposition fewer in Book 11 1., the Heronic proposition ill. 12 being no doubt omitted* In speaking of particular propositions, I shall use Heiberg's numbering, except where otherwise stated. The difference of 10 propositions between Thabit*Ishaq and at-Tusi 15 accounted for thus : (i) The three propositions Vl 12 and x. 28, 29 which both Ishaq and the Greek text have are omitted in at-Tusi. (2) Ishaq divides each of the propositions Xlll. i — 3 into two, making six instead of three in at-Tus! and in the GreeL (3) Ishaq has four propositions (numbered by him viii. 24^ 25, IX. 30. 31) which are neither in the Greek Euclid nor in at*TusL Apart from the above differences aUHajjaj {so far as we know), Ishaq and at-Tusl agree \ but their Euclid shows many differences from our Greek text. These differences we will classify as follows ^ K Propositions. The Arabian Euclid omits VI L 20, 22 of Gregory's and August's editions (Heiberg, App. to Vol Ih pp. 428«32) ; VIIL 16, 17 ; X. 7, 8, 13, 16, 24, 112, 113, 114, besides a lemma vtdgo X. 13, the proposition X. 117 of Gregory's edition, and the scholium at the end of the Book (see for these Heiberg*s Appendix to Vol. in, pp. 382, 408 — 416); XL 38 in Gregory- and August (Heibcrg, App. to Vol, Iv, p. 354); XII. 6, 13, 14; {also all but the first third of Book XV.). The Arabian Euclid makes II L 11, 12 into one proposition, and divides some propositions {X. 31, 32 ; XL 31, 34; xlll. i — 3) into two each. The order is also changed in the Arabic to the following extent V, 12, 13 are interchanged and the order in Books VI., VIL, IX. — xiiL is : VL 1—8, 13, II, 12. % 10, 14— *7» ift 20, 18, 21, 22| 24, 26, 23, 25,27—30,32,31,33. VI I. I— 2Q, 22, 21, 23—28, 31, 32i 29, 30, 33 — 39. IX. 1-^13, 20, 14 — 19, 21 — 25, 27, 26, 28 — 36, with two new pro- positions coming before prop. 30. X. 1—6, 9—12, IS, 14, 17—23, 26—28, 25, 29—30, 31, 32, 33— 111,115. XL I— 30, 3i> 32, 34, 33, 35— 39. XIL I— 5, 7, 9, 8, 10, 12, II, IS, 16—18. XIIL 1—3, s, 4, 6, 7, 12, 9, 10, 8, II, 13, IS, 14, 16—18. > See Klamroth, pp. 975—6, 38o» 383—4, 314— > 5* 3^6 ; Hdbeig, toL v. pp. xcvi, xcvii. k CH.VII] EUCLID IN ARABIA 8i 2. Definitions, The Arabic omits the following definitions: iv. Deff. 3 — 7, vil. Def. 9 (or 10), XI. Deff. 5 — 7, 15, 17, 23, 25 — 28; but it has the spurious definitions VI. Deff. 2, 5, and those of proportion and ordered proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges V. Deff. II, 12 and also vi. Deff. 3, 4. The order of the definitions is also different in Book vil. where, after Def 11, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23, and in Book XI. where the order is i, 2, 3,4, 8. 10, 9, 13, 14, 16, 12, 21, 22, 18, 19, 20, II, 24. 3. Lemmas andporisms. All are omitted in the Arabic except the porisms to vi. 8, VIII. 2, X. 3 ; but there are slight additions here and there, not found in the Greek, e.g. in vill. 14. 15 (in K). 4. Alternative proofs. These are all omitted in the Arabic, except that in X. 105, 106 they are substituted for the genuine proofs; but one or two alternative proofs are peculiar to the Arabic (Vl. 32 and vili. 4, 6). The analyses and syntheses to xill. i — 5 are also omitted in the Arabic. Klamroth is inclined, on a consideration of all these differences, to give preference to the Arabian tradition over the Greek (i) "on historical grounds," subject to the proviso that no Greek MS. as ancient as the 8th c. is found to contradict his conclusions, which are based generally (2) on the improbability that the Arabs would have omitted so much if they had found it in their Greek MSS., it being clear from the Fihrist that the Arabs had already shown an anxiety for a pure text, and that the old translators were subjected in this matter to the check of public criticism. Against the " historical grounds," Heiberg is able to bring a considerable amount of evidenced First of all there is the British Museum palimpsest (L) of the 7th or the b^inning of the 8th c. This has fragments of propositions in Book x. which are omitted in the Arabic; the numbering of one proposition, which agrees with the numbering in other Greek MS., is not comprehensible on the assumption that eight preceding propositions were omitted in it, as they are in the Arabic ; and lastly, the readings in L are tolerably like those of our MSS., and surprisingly like those of B. It is also to be noted that, although P dates from the loth c. only, it contains, according to all appearance, an ante-Theonine recension. Moreover there is positive evidence against certain omissions by the Arabians. At-Tusi omits VI. 12, but it is scarcely possible that, if Eutocius had not had it, he would have quoted VI. 23 by that number*. This quotation of VI. 23 by Eutocius also tells against Ishaq who has the proposition as VI. 25. Again, Simplicius quotes VI. 10 by that number, whereas it is VI. 13 in Ishaq ; and Pappus quotes, by number, XIII. 2 (Ishaq 3, 4), XIII. 4 (Ishaq 8), XlII. 16 (Ishaq 19). 1 Heiberg in Zeitschrift fur Math, u, PhysiJk, XXIX., hist.-litt Abth. p. asqq. ' Apollonius, ed. Heibeig, vol. 11. p. 318, 3 — 5. H. E. 6 it INTRODUCTION On the other hand the contraction of IIL n, 12 into one propositton in the Arabic tells in favour of the Arabic. Further, the omission of certain porisms in the Arabic cannot be supported; for Pappus quotes the porism to Xlll. 17*, P rod us those to I [. 4, III, I, VI L 2\ and Simplicius that to iv, 15. Lastly, some propositions omitted in the Arabic are required in later propositions. Thus X. 13 is used in X, 18, 22, 23, 26 etc. ; X. 17 is wanted in x, 18, 26^ 36; Xlh 6, 13 are required for XIL 11 and XIL 15 respectively. It must also be remembered that some of the things which were properly omitted by the Arabians are omitted or marked as doubtful in Greek MSS, also, especially in P, and others are rightly suspected for other reasons (ag. a number of alternative proofs, lenr^mas, and porisms, as well as the analyses and syntheses of Xlil. i — 5). On the other hand, the Arabic has certain interpolations peculiar to our inferior MSS, (cf the definition VL Def 2 and those of fitafiorlian and ordered proportion). Heiberg comes to the general conclusion that, not only is the Arabic tradition not to be preferred offhand to that of the Greek MSs., but it must be regarded as inferior in authority. It is a question how far the differences shown in the Arabic are due to the use of Greek MSS, differing from those which have been most used as the basis of our text, and how far to the arbitrary changes made by the Arabians themselves. Changes of order and arbitrary omissions could not surprise us, in view of the preface above quoted from the Oxford MS, of Thabit-Ishaq, with its allusion to the many important and necessary things left out by Abu *1 Wafa and to the author's own rearrangement of Books XIL, XIII. But there is evidence of differences due to the use by the Arabs of other Greek MSS. Heiberg' is able to show considerable resemblances between the Arabic text and the Bologna MS, b in that part of the MS. where it diverges so remarkably from our other MSS. (see the short description of it above, p, 49); in illustration he gives a comparison of the proofs of xtt, 7 in b and in the Arabic respectively, and points to the omission in both of the proposition given in Gregozy's edition as XI, 3S, and to a remark- able agreement t>etween them as r^ards the order of the propositions of Book XIL As above stated, the remarkable divergence of b only affects Books xi. (at end) and xii. ; and Book xiii. in b shows none of the transpositions and other peculiarities of the Arabic There are many differences between b and the Arabic^ especially in the definitions of Book XI., as well as in Book XIII. It is therefore a question whether the Arabians made arbitrary changes, or the Arabic form is the more ancient, and b has been altered through contact with other MSS. Heiberg points out that the Arabians must be sdone responsible for tlieir definition of a prism, which only covers a prism with a triangular base. This could not have been Euclid's own, for the word prism already has the wider meaning in Archimedes, and * Pappus, V. p. 436, 5. • Prodiu, pp. 303 — ^4. * ZeUsckr^fiir Maih.m. Pkysik^ XXIX., bist.-litt. Abth. p. 6aqq. , i IF ' CH. vii] EUCLID IN ARABIA 83 Euclid himself speaks of prisms with parallelograms and polygons as bases (xi. 39 ; XII. 10). Moreover, a Greek would not have been likely to leave out the definitions of the " Platonic " regular solids. Heiberg considers that the Arabian translator had before him a MS. which was related to b, but diverged still further from the rest of our MSS. He does not think that there is evidence of the existence of a redaction of Books I. — x. similar to that of Books XL, XII. in b ; for Klamroth observes that it is the Books on solid geometry (xi. — XIII.) which are more remarkable than the others for omissions and shorter proofs, and it is a noteworthy coincidence that it is just in these Books that we have a divergent text in b. An advantage in the Arabic version is the omission of VII. DeC 10, although, as lamblichus had it, it may have been deliberately omitted by the Arabic translator. Another advantage is the omission of the analyses and syntheses of xill. i — 5 ; but again these may have been omitted purposely, as were evidently a number of porisms which are really necessary. One or two remarks may be added about the Arabic versions as compared with one another. Al-Hajjaj*s object seems to have been less to give a faithful reflection of the original than to write a useful and convenient mathematical text-book. One characteristic of it is the careful references to earlier propositions when their results are used. ' Such specific quotations of earlier propositions are rare in Euclid ; but in al-Hajjaj we find not only such phrases as "by prop, so and so," " which was proved " or " which we showed how to do in prop, so and so," but also still longer phrases. Sometimes he repeats a construction, as in 1. 44 where, instead of constructing "the parallelo- gram BEFG equal to the triangle C in the angle EBG which is equal to the angle Z>" and placing it in a certain position, he produces AB to G, making BG equal to half DE (the base of the triangle CDE in his figure), and on GB so constructs the parallelogram BHKG by I. 42 that it is equal to the triangle CDE, and its angle GBH is equal to the given angle. Secondly, al-Hajjaj, in the arithmetical books, in the theory of proportion, in the applications of the Pythagorean 1. 47, and generally where possible, illustrates the proofs by numerical examples. It is true, observes Klamroth, that these examples are not apparently separated from the commentary of an-NairIzi, and might not there- fore have been due to al-Hajjaj himself; but the marginal notes to the Hebrew translation in Munich MS. 36 show that these additions were in the copy of al-Hajjaj used by the translator, for they expressly give these proofs in numbers as variants taken from al-Hajjaj ^ These characteristics, together with al-Hajjaj*s freer formulation of the propositions and expansion of the proofs, constitute an in- telligible reason why Ishaq should have undertaken a fresh translation from the Greek. Klamroth calls Ishaq's version a model of a good translation of a mathematical text ; the introductory and transitional 1 Kkmroth, p. 310; Steinschneider, pp. 85 — 6. 6 — 2 84 INTRODUCTION *^ [ch. vu phrases are stereotyped and few in number^ the technical terms are simply and consistently rendered, and the less formal expressions connect themselves as closely with the Greek as is consistent with intelligibility and the character of the Arabic language. Only in isolat^ cases does the formulation of definitions and enunciations differ to any considerable extent from the original. In general, his object seems to have been to get rid of difficulties and unevennesses in the Greek text by neat devices, while at the same time giving a faithful reproduction of it^ There are curious points of contact between the versions of al-Iiajjaj and Thabit-Ishaq. For example, the definitions and enunciations of propositions are often word for word the same» Presumably this is owing to the fact that Ishaq found these de- finitions and enunciations already established in the schools in his time, where they would no doubt be learnt by heart, and refrained from translating them afresh, merely adopting the older version with some changes*. Secondly, there is remarkable agreement between the Arabic versions as regards the figures, which show considerable variations from the figures of the Greek text, especially as regards the letters ; this is also probably to be explained in the same way, all the later translators having most likely borrowed al^Hajjaj's adaptation of the Greek figures'. Lastly, it is remarkable that the version of Books XL — xni, in the Kjjrfbenhavn Ms. (K), purporting to be by al-Hajjaj, is almost exactly the same as the Thabtt- Ishaq version of the same Books in O. Klamroth conjectures that Ishaq may not have translated the Books on solid geometry at all, and that Thabit took them from al-yajjaj, only making some changes in order to fit them to the translation of Ishaq*. From the facts (i) that at-Tusi's edition had the same number of propositions (468) as al-Hajjaj's version, while Thabit- Ishaq's had 478, and (2) that at-Tusi has the same careful references to earlier propositions, Klamroth concludes that at-TusI deliberately preferred al-^ajjaj's version to that of Ishaq*. Heiberg, however, points out (i) that at-TusI left out Vl iz which, if we may judge by Klamroth's silence, al-I;^ajjaj had, and (2) al-Hajjaj's version had one proposition less in Books L and IlL than at-Tusi has. Besides^ in a passage quoted by I:Iaji Khalfa* from at-TusI, the latter says that ** he separated the | things which, in the approved editions, were taken from the archetype j from the things which had been added thereto," indicating that he ; had compiled his edition from dotA the earlier translations'. i There were a large number of Arabian commentaries on, or ; reproductions of, the Elements or portions thereof, which will be 1 1 Klamroth, p. 390, illnstrates Is^*s method hy hit way of distiiigaifhin^ i^aptnb^ i (to be oongnient with) and 4^apiUfwBai fto be applied to), the confusion of which by timnt- J laton was animadverted on by Savile. uStjIkxi avoided the confusion by using two entirely < different words. ^ ' Klamroth, pp. 310 — i. * ibid, p. aSy. ' ^ iHd, pp. 304 — 5. * Uid. p. 374. • tMd, pp. 304—5. • miKhia&,i. p. 383. ' Hoberg, Zeitsckrififir Math. m. Pkysik^ xxix., hist-litt. Abth. pp. a, 3. CH. vii] EUCLID IN ARABIA 85 found fully noticed by Steinschneider^ I shall mention here the commentators etc. referred to in the Fihrist, with a few others. 1. Abu '1 'Abbas al-Fadl b. Hatim an-NairlzI (bom at Nairiz, died about 922) has already been mentioned*. His commentary survives, as r^^rds Books I. — vi., in the Codex Leidensis 399, i, now in course of publication by Besthorn and Heiberg, and as regards Books I. — ^X. in the Latin translation made by Gherard of Cremona in the nth c. and now published by Curtze from a Cracow MS.' Its importance lies mainly in the quotations from Heron and Simplicius. 2. Ahmad b. 'Umar al-Kar&blsI (date uncertain, probably 9th — loth c), ''who was among the most distinguished geometers and arithmeticians*." 3. A1-' Abbas b. Sa'ld al-Jauharl (fl. 830) was one of the astro- nomical observers under al-Ma*mun, but devoted himself mostly to geometry. He wrote a commentary to the whole of the Elements^ from the beginning to the end ; also the " Book of the propositions which he added to the first book of Euclid*." 4. Muh. b. 'Isa Abu 'Abdallah al-Mahtal (d. between 874 and 884) wrote, according to the Fihrist, (i) a commentary on Eucl. Book v., (2) " On proportion," (3) " On the 26 propositions of the first Book of Euclid which are proved without reductio ad absurdum^' The work " On proportion " survives and is probably identical with, or part of, the commentary on Book v.' He also wrote, what is not mentioned by the Fihrht^ a commentary on Eucl. Book X., a fragment of which survives in a Paris MS.* 5. Abu Ja'far al-Khftzin (i.e. " the treasurer " or '* librarian "), one of the first mathematicians and astronomers of his time, was born in Khurasan and died between the years 961 and 971. The Fikrist speaks of him as having written a commentary on the whole of the Elements*^ but only the commentary on the beginning of Book x. survives (in Leiden, Berlin and Paris) ; therefore either the notes on the rest of the Books have perished, or the Fikrist is in error *•. The latter would seem more probable, for, at the end of his commentary, al-Khazin remarks that the rest had already been commented on by Sulaiman b. 'Usma (Leiden MS.)" or 'Oqba (Suter), to be mentioned below. Al-Khazin's method is criticised unfavourably in the preface to the Oxford MS. quoted by Nicoll (see p. y7 above). 6. Abii '1 Wafa al-BuzjanI (940-997), one of the greatest Arabian mathematicians, wrote a commentary on the Elements, but ^ Steinschneider, Zeitschriftfur Math, u, Physik^ xxxi., hist.-litt. Abth. pp. 86 sqq. ' Steinschneider, p. 86, Pihrist (tr. Suter), pp. 16, 67 ; Suter, Die Mathematiier und AHronomen tier Araber (1900), p. 45. ' Supplementum ad Eucltdis opera omnia^ ed. Heiberg and Menge, Leipzig, 1899. * Fihrist^ pp. 16, 38 ; Steinschneider, p. 87 ; Suter, p. 65. * Fihrisi^ pp. 16, 35; Steinschneider, p. 88; Suter, p. n. * Fikrist, pp. 16, 25, 58. ' Suter, p. 36, note, quotes the Paris MS. 3467, 16** containing the work "on proportion'* as the authority for this conjecture. * MS. 3457, 39<» (cf. Woepcke in Mhn, pris, h Facad. des sciinces, xiv., 1856, p. 669). * Fikrist, p. 17. *• Suter, p. 58, note b. " Steinschneider, p. 89. 86 INTKODUCnOH ^micu* Vtt I ^! did not complete it*. His method is also unfavourably iri^rded in I the same preface to the Oxford MS. 28a According to Haji Khalfa, he also wrote a book on geometrical constructions, in thirteen chapters. Apparently a book answering to this description was compiled by a gifted pupil from lectures by Abu 1 Wafa, and a Paris MS. (Anc fends 169) contains a Persian translation of this work^ not that of Abu '1 Wafa himself. An analysis of the work was given by Wocpcke', and some particulars will be found in Cantor'* Abu '1 Wafa also wrote a commentary on Diophantus, as well as a separate " book of proofs to the propositions which Diophantus used in his book and to what he (Abu '1 Wafa) employed in his commentary*/' 7. Ibn Rfthawaihl al-ArjinT also commented on EucI, Book X,* 8. 'All b. Ahmad Abu U-Qasim al-AntikI (d. 987) wrote a commentary on the whole book*; part of it seems to survive (from the 5th Book onwards) at Oxford (CataL MSS. orient. II. 281 )^ 9. Sind b. *Ali Abu *t-Taiyib was a Jew who went over to Islam in the time of aKMa'niun, and was received among his astro- nomical observers, whose head he became" (about 830) ; he died after 864. He wrote a commentary on the whole of the EkmtHts ; " Abu *AlI saw nine books of it^ and a part of the tenthV His book " On the Apotomae and the Medials " mentioned by the Fikrisi^ may be the same as, or part of, his commentary on Book X* 10. Abu Yusuf Ya'qub b. Muh. ar-Razf '^ wrote a commentafy on Book X., and that an excellent one, at the instance of Ibn al- *AmId"." 11. The Fihrisi next mentions al> Kindt (Abu Yusuf Ya'qQb b. Ishaq b. as-Sabbah al-Kindi, d. about 873), as the author (1) of a work " on the objects of Euclid's book/' in which occurs the statement that the Elements were originally written by Apollonius, the carpenter (see above, p. 5 and note), (2) of a book ''on the improvement of Euclid's work," and (3) of another " on the improvement of the 14th and 15th Books of Euclid." ''He was the most distinguished man of his time, and stood alone in the knowledge of the old sciences collectively ; he was called ' the philosopher of the Arabians ' ; his writings treat of the most different branches of knowledge, as logic, philosophy, geometry, calculation, arithmetic, music, astronomy and others"." Among the other geometrical works of al-Kindi mentioned \ by the Fihrisi^ are treatises on the closer investigation of the results ! of Archimedes concerning the measure of the diameter of a circle in ^' terms of its circumference, on the construction of the figure of the two \ mean proportionals, on the approximate determination of the chords 1 • Fikristt p. 17. I • WoepdLcyimmaiAnaH^tte, Sit. v. T. v. pp. «i8— 956 and 300—350. l • Geuk. d. Math, vol. ig, pp. 743—6. « mhriti^ p. 39 ; Suter, p. 71. • Rkrisi, p. 17 ; Sater, p. 17. • Rkrisi^ p. 17. f Suter, p. 64. • Fihrisi^ p. 17, 39 ; Suter, pp. 13, 14. • Hhri$t, p. 17. M Fihrist^ p. 17 ; Suter, p. 66. » FihrisU p. 17, ia-15. ^ The mere catalogue of al-Kindl't works on the various boAclies of idence takes up Ibpr octavo pages (i i— 15) of Suter's translation of the Ftkrist. 1 CH. vii] EUCLID IN ARABIA 87 of the circle, on the approximate determination of the chord (side) of the nonagon, on the division of triangles and quadrilaterals and con- structions for that purpose, on the manner of construction of a circle which is equal to the surface of a given cylinder, on the division of the circle, in three chapters etc. 12. The physician Nazlf b. Yumn (or Yaman) al-Qass ("the priest") is mentioned by the Fihrist as having seen a Greek copy of Eucl. Book X. which had 40 more propositions than that which was in general circulation (containing 109), and having determined to translate it into Arabic^ Fragments of such a translation exist at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab, in Woepcke's tract); No. 18 contains "additions to some propositions of the loth Book, existing in the Greek language"." Nazlf must have died about 990'. 13. Yu^iannft b. Yusuf b. al-Iiarith b. al-Bitnq al-Qass (d. about 980) lectured on the Elements and other geometrical books, made translations from the Greek, and wrote a tract on the " proof" of the case of two straight lines both meeting a third and making with it, on one side, two angles together less than two right angles*. Nothing of his appears to survive, except that a tract "on rational and irrational ^ magnitudes," No. 48 in the Paris MS. just mentioned, is attributed ' to hinL I 14. Abu Muh. al-Hasan b. 'Ubaidallah b. Sulaiman b. Wahb . (d. 901) was a geometer of distinction, who wrote works under the y two distinct titles " A commentary on the difficult parts of the work r of Euclid " and " The Book on Proportion*." Suter thinks that an- other reading is possible in the case of the second title, and that it may refer to the Euclidean work "on the divisions (of figures)*." , 15. Qustft b. Luq§ al-Ba'labakki (d. about 912), a physician, philosopher, astronomer, mathematician and translator, wrote " on the difficult passages of Euclid's book" and "on the solution of arith- metical problems from the third book of Euclid'"; also an "intro- duction to geometry," in the form of question and answer". 16. Thabit b. Qurra (826-901), besides translating some parts of Archimedes and Books V. — vii. of the Conies of Apollonius, and revising Ishaq*s translation of Euclid's Elements^ also revised the trans- lation of the Data by the same Ishaq and the book On divisions of figures translated by an anonymous writer. We are told also that he wrote the following works: (i) On the Premisses (Axioms, Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On the propositions and questions which arise when two straight lines are cut by a third (or on the " proof" of Euclid's famous postulate). The last tract is extant in the MS. discovered by Woepcke (Paris 2457, 32®). He is also credited with "an excellent work" in the shape of an " Introduction to the Book of Euclid," a treatise on * Fikristf pp. 16, 17. " Woepcke, Mhn.pris. h Vacad, des sdenut^ XI v. pp. 666, 668. » Suter, p. 68. * Fihnst, p. 38 ; Suter, p. • Fihrist^ p. «6, and Suter's note, p. 60. • Suter, p. an, note 13. ' Fihrist, p. 43. • Fihrist^ p. 43 ; Suter, p. 60. 41. wm 88 INTRODUCTION [ch. vii I Geometry dedicated to fsma^n b. Bulbul, a Compendium of Geometry, and a large number of other works for the titles of which reference may be made to Sutcr, who also gives particulars as to which are extant^ 17. Abu Sa'ld Sinin b* Thabit b, Qurra, the son of the translator of Euclid, followed in his father's footsteps as geometer, astronomer and physician. He wrote an ''improvement of the book of ,.-,.. on the Elements of Geometry, in which he made various additions to the original" It is natural to conjecture that Emlid is the name missing in 3iis description (by Ibn abl Usaibi'a); Casiri has the name Aqaton*. The latest eaitor of the Tdrlkh al-Hukantd^ however^ makes the name to be Iflaton (« Plato), and he refers to the statement by the Fihrist and Ibn" al-QiftI attributing to Plato a work on the Elements of Geometry translated by Qusta. It is just possible, therefore, that at the time of Qusta the Arabs were acquainted with a book on the Elements of Geometry translated from the Greek, which they attri- buted to Plato*. Sinan died in 943. 18. Abu Sahl Wijan (or Waijan) b. Rustam al-Kuhl (fl, 988), bom at Kuh in Tabaristan, a distinguished geometer and astronomer, wrote, according to the Fihrist, a " Book of the Elements " after that of Euclid^; the ist and 2nd Books survive at Cairo, and a part of the 3rd Book at Berlin {^<^22Y, He wrote also a number of other geometrical works: Additions to the 2nd Book of Archimedes on . the Sphere and Cylinder (extant at Paris, at Leiden* and in the India ( Office), On the finding of the side of a heptagon in a circle (India Office and Cairo), On two mean proportionals (India OfficeX which last may be only a part of the Additions to Archimedes' On the Sphere -% and Cylinder, etc '\ 19. Abu Nasr Muh. b. Muh. b. Tarkhan b. Uzlag al-FtrftbT (870-950) wrote a commentary on the difficulties of the introductory matter to Books I. and v.* This appears to survive in the Hebrew translation which is, with probability, attributed to Moses b. Tibbon^ i 2a Abu 'All al-^asan b. al-Hasan b. al-Haitham (about 965- ] 1039), known by the name Ibn al-Haitham or Abu 'All al-Basrl, was a man of great powers and knowledge, and no one of his time approached I him in the field of mathematical science. He wrote several works on Euclid the titles of which, as translated by Woepcke from Usaibi'a, i are as follows" 1. Commentary and abridgment of the Elements. i 2. Collection of the Elements of Geometry and Arithmetic, f drawn from the treatises of Euclid and ApoUonius. j 3. Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements. \ * Sater, pp. 54—8. * Purist (ed. Suter), p. 59, note 13a ; Suter, p. 51, note b. ■ See Suter in Bibhatkeca Matkematica, iv„ 1903-4, pp. 196—7, reriew of Julias Lippert's Km ai-Qifii. Ta'ruh al-kukamd, Leiprig, 1903. * Fikristt p. 40. • Suter, p. 75. * Suter, p. 55. f SteioKhneider, p. 9«. * Stcintcnncider, pp. 93 — ^3. I r i 1 CH. vii] EUCLID IN ARABIA 89 4. Treatise on "measure" after the manner of Euclid's Elements. 5. Memoir on the solution of the difficulties in Book i. 6. Memoir for the solution of a doubt about Euclid, relative to Book V. 7. Memoir on the solution of a doubt about the stereometric portion. 8. Memoir on the solution of a doubt about Book Xil. 9. Memoir on the division of the two magnitudes mentioned in X. I (the theorem of exhaustion). ID. Commentary on the definitions in the work of Euclid (where Steinschneider thinks that some more general expression should be substituted for " definitions "). The last-named work (which Suter calls a commentary on the Postulates of Euclid) survives in an Oxford MS. (CataL MSS. orient. I. 908) and in Algiers (1446, i®). A Leiden MS. (966) contains his Commentary "on the difficult places " up to Book v. We do not know whether in this commentary, which the author intended to form, with the commentary on the Musadarat, a sort of complete commentary, he had collected the separate memoirs on certain doubts and difficult passages mentioned in the above list. A commentary on Book v. and following Books found in a Bodleian MS. (Catal. II. p. 262) with the title " Commentary on Euclid and solution of his difficulties " is attributed to b. Haitham ; this might be a continuation of the Leiden MS. The memoir on x. i appears to survive at St Petersburg, Ma de rinstitut des langues orient 192, 5* (Rosen, Catal. p. 125). 21. Ibn SInft, known as Avicenna (980-1037), wrote a Com- pendium of Euclid, preserved in a Leiden MS. No. 1445, and forming the geometrical portion of an encyclopaedic work embracing Logic, Mathematics, Physics and Metaphysics^ 22. Ahmad b. al-Husain al-AhwftzT al-Katib wrote a com- mentary on Book X., a fragment of which (some 10 pages) is to be found at Leiden (970), Berlin (5923) and Paris (2467, i8'*)*. 23. Naslraddin at-Tusi (i 201-1274) who, as we have seen, brought out a Euclid in two forms, wrote: 1. A treatise on the postulates of Euclid (Paris, 2467, 5®). 2. A treatise on the 5th postulate, perhaps only a part of the foregoing (Berlin, 5942, Paris, 2467, 6®). 3. Principles of Geometry taken from Euclid, perhaps identical with No. i above (Florence, Pal. 298). 4. 105 problems out of the Elements (Cairo). He also edited the Data (Berlin, Florence, Oxford etc.)'. 24. Muh. b. Ashraf Shamsaddin as-Samarqandl (fl. 1276) wrote " Fundamental Propositions, being elucidations of 35 selected proposi- ^ Steinschneider, p. 91 ; Suter, p. 89. ' Suter, p. 57. • Suter, pp. 150 — I. u ^a 90 INTRODUCTION [cm. vii tions of the first Books of Euclid " which are extant at Gotha (1496 and 1497), Oxford (Catal l. 967, 2^), ^^^ Brit Uus.\ 25, Musa b- Muh. b* Mahmud, known as Qa^izade ar-Ruml (i,c» the son of the judge from Asia Minor), who died between 1436 and 1446, wrote a commentary on the "Fundamental Propositions" ju^t mentioned^ of which many MSS* are extant'. It contained biographical statements about Euclid alluded to above (p. S, note), 26, Abu Da'ud Sulaiman b, 'Uqba^ a contemporary of aKKhazin (see above, No* 5), wrote a commentary on the second half of Book x., which is, at least partly, extant at Leiden (974) under the title " On the binomials and apotomae found in the lOth Book of Euclid V 27, The Codex Letdensis 399^ i containing al-Hajjaj*s transla- tion of Books L — VL is said to contain glosses to it by Sa'id b* Mas'ud b. al-Qass, apparently identical with Abu Nasr Gars al-Na'ma, son of the physician Mas'^Od b. al-Qass al-Bagdadi, who lived in the time of the last Caliph al-Musta'sini (d. 1258)*, 28, Abu Muhammad b. Abdalbiql al-Bagdadt al*Fara^i (d* 1 141, at the age of over 70 years) is stated in the Tdrtkh al-Hukamd to have written an excellent commentary on Book x. of the Elements, in which he gave numerical examples of the propositionsV This is published in Curtze's edition of an-Nainzi where it occupies pages 252 — 386* 29, Yahya b, Muh. b, 'Abdan b. *Abdalwahid, known by the name of Ibn al-Lubudl (1210-126S), wrote a Compendium of Euclid^ and a short presentation of the postulates'. 30, Abu 'Abdallah Muh. b, Mu'adh al-Jayyanl wrote a com- mentary on Eucl, Book v. which survives at Algiers (1446, 3*)'. 31, Abu Nasr Mansur b* 'All b. 'Iraq wrote^ at the instance of Muh. b, Ahmad Abu V-Raihan al-Blruni (973-1048), a tract "on a doubtful (difficult) passage in Eucl Book XIII/' (Berlin, S925)'. ' Stit«T, p. 157- * *^*^' P" *75- * *^" P- 5**' < ihid. pp, [55— 4 i *V' ° GaiU, i^ i4 ; Suloschiieider, pp. 94 — 5. * Safer in BWioiMsca MaiAemaiua^ iv^, 1904, pp. «5» 195 ; Sater has also an artide on its contents, BiUioikica MaiJkimaiuat viu, 1906^71 pp. 354 — 351. ^ Steinschneider, p. 94 ; Sater, p. 14O. > Sater, Nacktrdge und Berickttgwtgen^ in Abkandimngen mr Gesck. der math, fVUsin- sehapen^ xiv., 1903, p. 17a * Snter, p. 81, ana Na€htrag$^ p. 173. CHAPTER VIII. PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS. Cicero is the first Latin author to mention Euclid^; but it is not likely that in Cicero's time Euclid had been translated into Latin or was studied to any considerable extent by the Romans; for, as Cicero says in another place', while geometry was held in high honour among the Greeks, so that nothing was more brilliant than their mathematicians, the Romans limited its scope by having regard only to its utility for measurements and calculations. How very little theoretical geometry satisfied the Roman agrimensares is evidenced by the work of Balbus de mensuris^^ where some of the definitions of Eucl. Book I. are given. Again, the extracts from the Elements found . in the fragment attributed to Censorinus (fl. 238 A.D )* are confined to I the definitions, postulates, and common notions. But by degrees the Elements passed even among the Romans into the curriculum of a ! liberal education ; for Martianus Capella speaks of the effect of the enunciation of the proposition "how to construct an equilateral triangle on a given straight line " among a company of philosophers, ' who, recognising the first proposition of the Elements, straightway break out into encomiums on Euclid*. But the Elements were then ; {c. 470 A.D.) doubtless read in Greek ; for what Martianus Capella gives* was drawn from a Greek source, as is shown by the occurrence of Greek words and by the wrong translation of I. def. i (" punctum vero est cuius pars nihil est"). Martianus may, it is true, have quoted, not from Euclid himself, but from Heron or some other ancient source. But it is clear from a certain palimpsest at Verona that some scholar had already attempted to translate the Elements into Latin. This palimpsest' has part of the " Moral reflections on the Book of Job " by Pope Gregory the Great written in a hand of the 9th c. above certain fragments which in the opinion of the best judges date from the 4th c Among these are fragments of Vergil and of Livy, as well as a geometrical fragment which purports to be taken from the 14th and 15th Books of Euclid. As a matter of fact it is from Books XIL and XIII. and is of the nature of a free rendering, or rather a new * Dearatort III. 13a. • Tusc, I. 5. * GromaHH veteres, I. 97 sq. (ed. F. Blume, K. Lachmann and A. Rudorff. Berlin, 1848, 1851). * Censorinus, ed. Hultsch, pp. 60— 3. * Martianus Capella, vi. 734. * ibid, vi. 708 sq. ^ Cf. Cantor, i„ p. 565. :; INTRODUCTION [ch, viii arrangement, of Euclid with the propositions in different orders The MS- was evidently the translator's own copy, because some words arc struck out and replaced by synonyms. We do not know whether the translator completed the translation of the whole* or in what relation his version stood to our other sources. Magnus Aurelius Cassiodorius (b* about 475 a.d.) in the geometrical part of his encyclopaedia Deartibus ac disciplinis Hberalium literarum says that geometry was represented among the Greeks by Euclid, Apollonius, Archimedes, and others, "of whom Euclid was given us translated into the Latin language by the same great man Boethius**; also in his collection of letters* is a letter from Theodoric to Boethius containing the words, "for in your translations ... Ntcomachus the arithmetician, and Euclid the geometer^ are heard in the Ausonian tongue" The so-called Geometry of Boethius which has come down to us by no means constitutes a translation of Euclid The MSS. variously give five, four, three or two Books, but they represent only two distinct compilations, one normally in five Books and the other in two. Even the latter, which was edited by Friedlein, is not genuine*, but appears to have been put together in the nth c», from various sources. It begins with the definitions of EucL I,, and in these are traces of perfectly correct readings which are not found even in the MSS. of the loth c, but which can be traced in Proclus and other ancient sources ; then come the Postulates (five only), the Axioms . (three only), and after these some definitions of Eucl 11., 11 L, iv. Next come the enunciations of Eucl I., of ten propositions of Book 11,, and of some from Books llL, IV., but always without proofs ; there follows an extraordinary passage which indicates that the author will now give something of his own in elucidation of Euclid, though what ' follows is a literal translation of the proofs of EucL I. I — 3, This latter passage, although it affords a strong argument against the genuineness of this part of the work, shows that the Pseudoboethius had a Latin translation of Euclid from which he extracted the three propositions. Curtze has reproduced, in the preface to his edition of the trans- lation by Gherard of Cremona of an-Nairizf s Arabic commentary on Euclid, some interesting fragments of a translation of Euclid taken from a Munich MS. of the loth c. They are on two leaves used for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 2*^ 757) atr>d consist of portions of Eucl, L 37, 38 and IL 8, translated literally word for word from the Greek text The translator seems to have been an Italian (c£ the words '' capitolo nono " used for the ninth prop, of Book II.) who knew very little Greek and had moreover little mathematical knowledge. For example, he translates the capital letters denoting points in figures as if they were numerals : thus tA ABF, * The frugment wms deciphered by W. Stndemmid, who oommimicated his resalts to Cantor. * CaiBodorius, Varioi^ i. 45, p. 40, 13 ed. Mommieii. * See etpedaUy Weissenbom in AbkatuUumgm mr Getck, d. Math. II. p. 185 iq.; 1 Heibcsg in Pkilol^gus^ XLUI. p. 507 sq. ; Cantor, ig, p. 580 sq. CH. viii] TRANSLATIONS AND EDITIONS 93 AEZ is translated ''que primo secundo et tertio quarto quinto et scptimo," T becomes "tricentissimo " and so on. The Greek MS. which he used was evidently written in uncials, for AEZ6 becomes in one place " quod autem septimo nono," showing that he mistook AE for the particle &', and koI 6 2TU is rendered ''sicut tricentissimo et quadringentissimo/' showing that the letters must have been written KAIOCTU. The date of the Englishman Athelhard (iEthelhard) is approxi- mately fixed by some remarks in his work Perdiffidles Quaestiones Naturales which, on the ground of the personal allusions they contain, must be assigned to the first thirty years of the 12th c* He wrote a number of philosophical works. Little is known about his life. He is said to have studied at Tours and Laon, and to have lectured at the latter school. He travelled to Spain, Greece, Asia Minor and Egypt, and acquired a knowledge of Arabic, which enabled him to translate from the Arabic into Latin, among other works, the Elements of Euclid. The date of this translation must be put at about 11 20. MSB. purporting to contain Athelhard's version are extant in the British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 47 and Ball. Coll. 257 of 12th c), Nurnberg (Johannes Regiomontanus' copy) and Erfurt Among the very numerous works of Gherard of Cremona (i 1 14 — 1 1 87) are mentioned translations of ** 15 Books of Euclid" and of the Data}. Till recently this translation of the Elements was supposed to be lost; but Axel Anthon Bjombo has succeeded (1904) in discovering a translation from the Arabic which is different from the two others known to us (those by Athelhard and Campanus respectively), and which he, on grounds apparently convincing, holds to be Gherard's. Already in 1901 Bjombo had found Books x. — xv. of this translation in a MS. at Rome (Codex R^inensis lat. 1268 of 14th c.)'; but three years later he had traced three MSB. containing the whole of the same translation at Paris (Cod. Paris. 7216, 15th c), Boulogne-sur-Mer (Cod. Bononiens. 196, 14th c), and Bruges (Cod. Brugens. 521, 14th c), and another at Oxford (Cod. Digby 174, end of 12th c.) containing a fragment, XI. 2 to XIV.* The occurrence of Greek words in tfiis translation such as rombus^ romboides (where Athelhard keeps the Arabic terms), ambligonitis, orthogonius, gnomo^ fyramis etc., show that the translation is independent of Athelhard's. Gherard appears to have had before him an old translation of Euclid from the Greek which Athelhard also often followed, especially in his terminology, using it however in a very different manner. Again, there are some Arabic terms, e.g. meguar for axis of rotation^ which Athelhard did not use, but which is found in almost all the translations that are with certainty attributed to Gherard of Cremona; there occurs also the » Cantor, Gesch, d. Math, i„ p. 006. * Boncompagni, Delia vita e delU cfere di Gherardo Crewunuse^ Rome, 1851, p. 5. * Described in an appendix to Studien iiber Menelaos^ Sphdrik {Abhanduungm %ur Gtschuhte der mathematischen IVissenschafien, xiv., 1901). * Sec Bibliotheca Mathematica, vig, 1905-^, pp. 141—8. 94 INTRODUCTION [ck. viii expression "superficies equidjstantium latcnim et rectomm angulorum/' found also in Gherard's translation of an-NairM, where Athelhard says "parallelogrammum rectan^lum/' The translation is much clearer than Athelhard's: it is neither abbreviated nor "edited'' as Athelhard's appears to have been ; it is a word-for*word translation of an Arabic MS. containing a revised and critical edition of Thabit^s version. It contains several notes quoted from Thabit himself {Thebit dixii\ e.g. about alternative proofs etc* which Thabit found ''in another Greek MS./* and is therefore a further testimony to Thabit s critical treatment of the text after Greek MSS. The new editor also added critical remarks of his own* e.g. on other proofs which he found in other Arabic versions, but not in the Greek; whence it is clear that he compared the Thabit version before him with other versions as care- fully as Thabit collated the Greek MSS. Lastly, the new editor speaks of ''Thebit qui transtulit hunc librum in arabicam linguam" and of "translatioThebit," which may tend to confirm the statement of al-Qifti who credited Thabit with an independent translation, and not (as the Fihrist does) with a mere improvement of the version of Ishaq b, ^unain. Gherard's translation of the Arabic commentary of an-NatiizI on the first ten Books of the EUments was discovered by Maximitian Curtze in a MS. at Cracow and published as a supplementary volume to Heiberg and Menge's Euclid^: it will often be referred to in this work. Next in chronological order comes Johannes Campanus (Cam pane) of Novara. He is mentioned by Roger Bacon (12(4-1294) as a prominent mathematician of his time*, and this indication of his date is confirmed by the fact that he was chaplain to Pope Urban IV, who was Pope from 1261 to 1281*. His most important achievement was his edition of the Elements including the two Books XIV. and XV. which are not Euclid's. The sources of Athelhard's and Campanus' translations, and the relation between them, have been the subject of much discussion, which does not seem to have led as yet to any definite conclusion. Cantor (III, p. 91) gives references^ and some particulars. It appears that there is a MS. at Munich (Cod. lat. Mon. 1 3021) written by Sigboto in the 12th c at Priifning near Regensburg, and denpted by Curtze by the letter R, which contains the enunciations of part of Euclid. The Munich MSS. of Athelhard and Campanus' translations have many enunciations textually identical with those in R, so that the source of all three must, for these enunciations, have ' AnariHi in decern HbrM friores EUmenUrum EucUdis Commentarii ex inter^niaUcmi Gherardi Cremonensis in codice Cracaviensi 569 servata edidit Maximiliaous Cunze, Leiptig (Teubner), 1899. * Cantor, Hi, p. 88. ' Tiraboschi, Storia della leiteratura ita/iana, IV. 145—160. * H. V^eissenbom in ZeUschrift fiir Math, «. Pkysik^ xxv., Supplement, pp. T43— 166, and in his monograph, Die Obenettungen des Ettkliddurck Campano und ZambertiXi^i) ; Max. Curtze in FhtloUgiiche Rundschau (1881), I. pp. 943^950, and in Jdhresbericht iiAtr die Forischritte der classischen AUerihumswissemchaft^ XL. (1C64, in.) pp. 19 — 39 ; Heibeig in ZaischriftJUr Math, u. Physik^ XXXV., hist.-litt Abth., pp. 48—58 and pp. 81—6. CH. viii] TRANSLATIONS AND EDITIONS 95 been the same; in others Athelhard and Campanus diverge com- pletely from R, which in these places follows the Greek text and is therefore genuine and authoritative. In the 32nd definition occurs the word " elinuam," the Arabic term for " rhombus," and throughout the translation are a number of Arabic figures. But R was not translated from the Arabic, as is shown by (among other things) its close resemblance to the translation from Euclid given on pp. 377 sqq. of the Gromatici Veteres and to the so-called geometry of Boethius. The explanation of the Arabic figures and the word " elinuam " in Def. 32 appears to be that R was a late copy of an earlier original with corruptions introduced in many places ; thus in Def. 32 a part of the text was completely lost and was supplied by some intelligent copyist who inserted the word "elinuam," which was known to him, and also the Arabic figures. Thus Athelhard certainly was not the first to translate Euclid into Latin ; there must have been in existence before the nth c. a Latin translation which was the common source of R, the passage in the Gromatici^ and '' Boethius." As in the two latter there occur the proofs as well as the enunciations of L i — 3, it is possible that this translation originally contained the proofs also. Athelhard must have had before him this translation of the enunciations, as well as the Arabic source from which he obtained his j proofs. That some sort of translation, or at least fragments of one, were available before Athelhard's time even in England is indicated by some old English verses^ : *'The clerk Euclide on this wyse hit fonde Thys craft of gemetry yn Egypte londe Yn Egypte he tawghte hyt ful wyde, In dyvers londe on every syde. Mony erys afterwarde y understonde Yer that the craft com ynto thys londe. Thys craft com into England, as y yow say, Yn tyme of good kyng Adelstone's day," which would put the introduction of Euclid into England as far back as 924-940 A.D. We now come to the relation between Athelhard and Campanus. That their translations were not independent, as Weissenborn would have us believe, is clear from the fact that in all MSS. and editions, apart from orthographical differences and such small differences as are bound to arise when MSS. are copied by persons with some knowledge of the subject-matter, the definitions, postulates, axioms, and the 364 enunciations are word for word identical in Athelhard and Campanus ; and this is the case not only where both have the same text as R but where they diverge from it. Hence it would seem that Campanus used Athelhard's translation and only developed the proofs by means of another redaction of the Arabian Euclid. It is true that the difference between the proofs of the propositions in the two translations is considerable; Athelhard's are short and com- ^ Quoted by Halliwell in Rara Maihematica (p. 56 note) from MS. Bib. Reg. Mas. Brit. 17 A. I. f. 2^—3. 96 INTltOmTCTICni pressed, Campanus' cleamr and mote c o mp lete, iolkMrlaff tile Gnei text more closely, thoadi still at aome diftaiioe» Ftetibtt^: lb arrangement in the two is diflerent; in AtbeUianl the praoft itgitM i ty | j precede the enunciation^ Dunpanos §oXkmm tlie mual ctdm: ft iaa question how far the differencet in the pnxA, and certain additiwiJtt each, are due to the two transtatoia tfaantdves or go back to AmUc originals. The latter supposition seems to Cocttt and Cantor the more probable one. Curtae's general view of tiie relation of Qininan n s to Athelhard is to the effect that Adidhard's tinnslalion was nwiMllj altered, from the form in iHiidi it appears in the two Emmt Wm. described by Weissenbom, fay socoesahre copyists and ciwiwi i mta tor s wAo had Arabic originmU m^itm tlmm^ w^ it^ took ti» faraa irilidi Campanus gave it and in which it was pnUished In soppoit of ttis view Curtze refers to Rq;iomontaniis*copy of the AtfadharcMranipanns translation. In R^omontanus' own prc&ce the title is given, 'Md this attributes the translation to Atfidhard; bit^ wiiite diia copy agrees almost exactly wiUi Athelhard In Book L, yet, in places wtoa Campanus is more lengthy, it has similar additions, and in the Isler Books, especially from Book Ul. onwards, agrees ahsdittd^ wfth Campanus; R^iomontanu% too^ himself implies tinrt, fhrnMh A translation was Athelhard's, Campanus had reidsed it; fcr ha !■ notes in the margin such as the following, **Campani est hec^" ''dnfako an demonstret hie Campanus " etc. * We come now to the printed editions of tlie udiole or of portfams of the ElemenU, This is not the idace for a complete bihlkiipaplqr» : such as Riccardi has attempted in his valuable memoir issued in five parts between 1887 and 1893, which makes a large book in itself ^ I shall confine myself to saying something of the most noteworthy translations and editions. It will be convenient to give first the Latin translations which preceded the publication of the editio prinups of the Greek text in 1 533, next the most important editions of the Greek text itself, and after them the most important translations arranged according to date of first appearance and languages, first the Latin translations after 1533, then the Italian, German, French and English translations in order. It may be added here that the first allusion, in the West, to the Greek text as still extant is found in Boccaccio's commentary on the Divina Commedia of Dante*. Next Johannes Regiomontanus, who intended to publish the Elements after the version of Campanus, but with the latter^s mistakes corrected, saw in Italy (doubtless when staying with his friend Bessarion) some Greek MSS. and noticed how far they differed from the Latin version (see a letter of his written in | the year 147 1 to Christian Roder of Hamburg)*. ^ Saggw di una Bibliogrt^ EucUdea^ memoria del Prof. Pietio Riccardi (Bologiui, ! 1887, 1888, 1890, 1893). ^ " I. p. 404. * Pabliahed in C. T. de Mail's fitmorabilia BibUatiucarum NorimUrgpumm^ Pttrt I. ' p. 190 iqq. i i CH. viii] TRANSLATIONS AND EDITIONS 97 I I. Latin translations prior to 1533. I 1482. In this year appeared the first printed edition of Euclid, which was also the first printed mathematical book of any import- ance. This was printed at Venice by Erhard Ratdolt and contained Campanus' translation \ Ratdolt belonged to a family of artists at Augsburg, where he was born about 1443. Having learnt the trade of printing at home, he went in 1475 to Venice, and founded there a famous printing house which he managed for 1 1 years, after which he returned to Augsburg and continued to print important books until 1 5 16. He is said to have died in 1528. Kastner' gives a short description of this first edition of Euclid and quotes the dedication to Prince Mocenigo of Venice which occupies the page opposite to the first page of text. The book has a margin of 2^ inches, and in this margin are placed the figures of the propositions. Ratdolt says in his dedication that at that time, although books by ancient and modern authors were printed every day in Venice, little or nothing mathematical had appeared : a fact which he puts down to the diffi- Iculty involved by the figures, which no one had up to that time succeeded in printing. He adds that after much labour he had discovered a method by which figures could be produced as easily as letters*. Experts are in doubt as to the nature of Ratdolt's discovery. Was it a method of making figures up out of separate parts of figures, straight or curved lines, put together as letters are put together to make words > In a life of Joh. Gottlob Immanuel Breitkopf, a con- temporary of Kastner's own, this member of the great house of Breitkopt is credited with this particular discovery. Experts in that ^same house expressed the opinion that Ratdolt's figures were wood- 'cuts, while the letters denoting points in the figures were like the other letters in the text ; yet it was with carved wooden blocks that printing b^an. If Ratdolt was the first to print geometrical figures, it was not long before an emulator arose ; for in the very same year IMattheus Cordonis of Windischgratz employed woodcut mathematical figures in printing Oresme*s De laHtudinibus\ How eagerly the opportunity of spreading geometrical knowledge was seized upon is proved by the number of editions which followed in the next few years. Even the year 1482 saw two forms of the book, though they only differ in the first sheet Another edition came out in i486 {Ulmae, apud lo, Regerutn) and another in 149 1 {Vincentiae per ^ Curtze (An-NairizI, p. xiii) reproduces the heading of the first page of the text as follows (there is no title-pa^e) : PreclarifTimu opus elemento^ Eucliois megarefis vna cu cOmentis Camponi pfpicacifnini in arte geometric incipit felicit*, after which the definitions begin at once. Other copies have the shorter heading : Preclarissimus liber elementonim Euclidis perspicacissimi : in artem Geometric incipit quam foelicissime. At the end stands I the following : C Opus elementoru euclidis megarenf is in geometria arte Jn id quoq} Campani I pfpicaciffinu C6mentationes finiut. Erhardus ratdolt Augustensis imprefTor folertiffimus . ; venetijs iropreirit . Anno ialutis . M.cccc.lxxxij . Octauis . Calefi . Jufi . Lector . Vale. t' Kastner, GtschUhte dtr Maihematik^ I. p. 189 sqq. See also Weissenbom, Die Obersett- ungen des EukHd durch Campano und Zamoertit pp. i — 7. ' "Mea industria non sine maximo labore efieci vt qua fturilitate litterarum elementa imprimuntur ea etiam geometrice figure conficerentur." * Curtze in Zeiischriftfiir Math, u, Physik, XX., hist.-litt. AbUi. p. 58. H. E. 7 I r 98 INTRODUCTION [cH.Tin Leonardum de BasUea ei GuHditmm d$ P^^i^\ but witiiout Ae dedi- cation to Mocenigo who had died in the meantime (1485)^ If Cam- panus added anything of his own, his additions are at all events not distinguished by any difference of t^pe or oAerwise; the enunciations are in large type, and the rest is prmted continuously in smaller type. There are no superscriptions to particular passages such as Ettduks ex Campano^ Campanus^ Cm^ami addiiio^ or Omi$mU amMtoHo^ which are found for the first time in the Paris edition of 1516 giving both Campanus' version and that of Zamberti Qwesently to be men- tioned). 1 501. G. Valla included in his enc3^opaedic work De expiUndis et fugiendis rebus published in this year at Venice (iVk eudibus Atdi Romant) a number of propositions with proofs and scholia translated from a Greek MS. which was once in hiis possession (cod Mutin. ni B, 4 of the isthc). 1 505. In this year Bartolomeo Zamberti (Zambertus) brought out at Venice the first translation, from the Greek text, of the whole of tlie Elements. From the titled as well sis from his prefaces to the CaUptriem and DaUif with their allusions to previous translators ** who take some things out of authors, omit some, and change some," or ** to that most barl^rous translator " who filled a volume purporting to be Euclid's ''with extraordinary scarecrows, nightmares and phantasies," one object of Zamberti's translation b dear. His animus against Campanus appears also in a number of notes, e.g. when he condemns the terms ''helmuain" and '^ helmuariphe "* used by Campanus as barbarous^ un-Latin etc, and when he is roused to wrath by Campanus' unfortu- I nate mistranslation of v. Def. 5. He does not seem to have had the > penetration to see that Campanus was translating from an Arabic, and not from a Greek, text Zamberti tells us that he spent seven years over his translation of the thirteen Books of the Elements. As he seems to have been bom in 1473, and the Elements were printed as early as 1500, though the complete work (including the Phaenomena^ Optica^ Catoptrica^ pata etc.) has the date 1505 at the end, he must have translated Euclid before the age of 3a Heiberg has not been able to identify the MS. of the Elements which Zamberti used ; but it is clear that it belonged to the worse class of MSS., since it contains most of the interpolations of the Theonine variety. Zam- berti, as his title shows, attributed th^ proofs to Theon. 1509. As a counterblast to Zamberti, Luca Paciuolo brought out an edition of Euclid, apparently at the expense of Ratdolt, at Venice f {per Pagamnum de Paganinii), in which he set himself to vindicate dampanus. The title-page of this now very rare edition' b^ns thus : | *'The works of Euclid of Megara, a most acute philosopher and without ' ^ The title begins thus: *'Eiiclidis megaresis philosophi platonicj mathematicaniml discipliiuurum Janitoris : Habent in hoc volumine quicunque ad matbematicaro substantiam . aspirant : elementomm libros xiij cum expositione Theonis insignis mathematici. quibus, multa quae deerant ex lectione graeca sumpta addita sunt nee non plarima peniena et praepostere: voluta in Campani interpretatione : ordinata digesta et castigata sant etc." For a description of the book see Weissenbom, p. is sqq. . ' See Weissenbom, p. 30 sqq. CH. viii] TRANSLATIONS AND EDITIONS 99 question the chief of all mathematicians, translated by Campanus their most faithful interpreter'' It proceeds to say that the translation had been, through the fault of copyists, so spoiled and deformed that it could scarcely be recognised as Euclid. Luca Paciuolo accordingly has polished and emended it with the most critical judgment, has corrected 129 figures wrongly drawn and added others, besides supply- ing short explanations of difficult passages. It is added that Scipio V^ius of Milan, distinguished for his knowledge '' of both languages'* (le. of course Latin and Greek), as well as in medicine and the more sublime studies, had helped to make the edition more perfect. Though Zamberti is not once mentioned, this latter remark must have refer- ence to Zamberti's statement that his translation was from the Greek text ; and no doubt Zamberti is aimed at in the wish of Paciuolo's " that others too would seek to acquire knowledge instead of merely showing off, or that they would not try to make a market of the things of which they are ignorant, as it were (selling) smoke*." Weissenborn observes that, while there are many trivialities in Paci- uolo's notes, they contain some useful and practical hints and explana- tions of terms, besides some new proofs which of course are not difficult if one takes the liberty, as Paciuolo does, of diverging from Euclid's order and assuming for the proof of a proposition results not arrived at till later. Two not inapt terms are used in this edition to describe the figures of ill. 7, 8, tihe former of which is called the gooses foot {pes anseris\ the second the peacock's tail {catula pavonis), Paciuolo as the castigator of Campanus' translation, as he calls himself, failed to correct the mistranslation of v. Def 5'. Before the fifth Book he inserted a discourse which he gave at Venice on the 15th August, 1508, in S. Bartholomew's Church, before a select audience of 500, as an introduction to his elucidation of that Book. 1 5 16. The first of the editions giving Campanus' and Zamberti's translations in conjunction was brought out at Paris (i« offidna Henrici Stephani e regione scholae Decretorum). The idea that only the enun- ciations were Euclid's, and that Campanus was the author of the proofs in his translation, while Theon was the author of the proofs in the Greek text, reappears in the title of this edition; and the enunciations of the added Books xiv., xv. are also attributed to Euclid, Hypsicles being credited with the proofs'. The date is not on the title-page nor at the ^ "Atque utinam et alii cognoscere vellent nod ostentare ant ea quae nesdunt velati fumum venditare non conarentar.'* ' Campanus' translation in Ratdolt's edition is as follows: "Quantitates quae dicnntur oontinuam habere proportionaliutem, sunt, quarum equ^ multiplicia aut equa sunt aut eou^ sibi sine interruptione addunt aut minuunt " ( !), to which Campanus adds the note : ^ Continue proportionalia sunt quorum omnia multiplicia equalia sunt continue proportionalia. Sed noluit ipsam diffinitionem proponere sub hac forma, quia tunc difiiniret idem per idem, aperte (? a parte) tamen rei est istud cum sua diffinitione convertibile." ' **£uclidis Megarensis Geometricorum Elementorum Libri xv. Campani Galli trans- alpini in eosdem commentariorum libri xv. Theonis Alexandrini Bartholomaeo Zambertd Veneto interprete, in tredecim priores, commentationum libri xiii. Hypsiclis Alexandrini in duos posteriores, eodem Bartholomaeo Zamberto Veneto interprete, commentariorum libri n." On the last page (161) is a similar sutement of content, but with the difference that the expression **ex Campani... deinde Theonis... et Hypsiclis... /nu/iM^t^Kr." For description see Weissenborn, p. 56 sqq. 7-^2 loo INTRODUCTION \ ' [€& vm end, but the letter of dedication to Franjois Briconnet by Jacques Leftvre is dated the day after the Epiphany, 151& The figures are in the margin. The arrangement of the propositions is as follows : first the enunciation with the beading Eudides ex Campano^ then the proof with the note Campanus^ and after that, as Campani additio^ any passage found in the ^ition of Campanus' translation but not in the Gr^k text ; then follows the text of the enunciation translated from the Greek with the heading Eudides ^x Zambtrto, and lastly the proof headed Theo ex ZambertQ, There are separate figures for the two proofs. This edition was reissued with few changes in 1537 and 1546 at Basel (fipud lokaumm H€rvagium\ but ivith the addition of the Pkojenontitta^ Optica^ Catoptrka etc For the edition of 1537 the Paris edition of 1516 was collated with *'a Greek copy" (as the preface says) by Christian Herlinj professor of mathematical studies at Strassburg, who however seems to have done no more than correct one or two passages by the help of the Basel editw princeps (1533), and add the Greek word in cases where Zamberti's translation of it seemed unsuit- able or inaccurate. ' We now come to # , t, . . T IL Editions of the Greek text, » •* 1 533 is the date of the editw prinaps, the title-page of which reads as follows: ETKAEIAOT STOIXEinN BIBA>- lEK EK TON eEONOZ 1TN0Y2I0R £«'? ToO avTov TO TrpmroVf e^^^dnop TlpoicXov 0ifi\* o* Adiecta praefatiuncula in qua de disctplinis # Mathematicis nonnihiL BASILEAE APVD lOAN, HERVAGIVM ANNO M^aXXXIlL MENSE SEPTEMBRL The editor was Simon Grynaeus the elder (d* 1541), who, after working at Vienna and Ofen, Heidelberg and Tubingen, taught last of all at Basel, where theology was his main subject. His "prae- fatiuncula" is addressed to an Englishman, Cuthbert Tonstall (1474- 1559)9 ^ho> having studied first at Oxford, then at Cambridge, where he became Doctor of Laws, and afterwards at Padua, where in addi- tion he learnt mathematics — mostly from the works of Regiomontanus and Paciuolo — wrote a book on arithmetic* as **a farewell to the sciences," and then, entering politics, became Bishop of London and member of the Privy Council, and afterwards (i 530) Bishop of Durham. Grynaeus tells us that he used two MSS. of the text of the Elements, entrusted to friends of his, one at Venice by " Lazarus Bayfius " (Lazare de BaTf, then the ambassador of the King of France at Venice), the other at Paris by " loann. Rvellius " (Jean Ruel, a French doctor and a Greek scholar), while the commentaries of Proclus were put at ^ De arti tufputamdiiiM qmatuor. 1 CH. viii] TRANSLATIONS AND EDITIONS ..;v loi the disposal of Grynaeus himself by "loann. Claymundus** j;t*Qxford. Heiberg has been able to identify the two MSS. used for* the** text ; they are (i) cod. Venetus Marcianus 301 and (2) cod. Paris. g^.*2343 of the 1 6th c, containing Books I. — XV., with some scholia wHicJ^* are embodied in the text. When Grynaeus notes in the margfff \t^e readings from "the other copy," this "other copy" is as a rule** the Paris MS., though sometimes the reading of the Paris MS. is taken, into the text and the " other copy " of the margin is the Venice Mfi* Besides these two MSS. Grynaeus consulted Zamberti, as is shown by^* a number of marginal notes referring to " Zampertus " or to " latinum ' exemplar" in certain propositions of Books IX. — XL When it is con- sidered that the two MSS. used by Grynaeus are among the worst, it is obvious how entirely unauthoritative is the text of the editio prinups. Yet it remained the source and foundation of later editions of the Greek text for a long period, the editions which followed being designed, not for the purpose of giving, from other MSS., a text more nearly representing what Euclid himself wrote, but of supplying a handy compendium to students at a moderate price. 1536. Orontius Finaeus (Oronce Fine) published at Paris {apud Simonetn Colinaeuni) "demonstrations on the first six books of Euclid's elements of geometry," " in which the Greek text of Euclid himself is inserted in its proper places, with the Latin translation of Barth. Zamberti of Venice," which seems to imply that only the enunciations were given in Greek. The preface, from which Kastner quotes S says that the University of Paris at tfiat time required, from all who aspired to the laurels of philosophy, a most solemn oath that they had attended lectures on the said first six Books. Other editions of Fine's work followed in 1544 and 1551. 1545. The enunciations of the fifteen Books were published in Greek, with an Italian translation by Angelo Caiani, at Rome {apud Antonium Bladum Asulanum), The translator claims to have cor- rected the books and " purged them of six hundred things which did not seem to savour of the almost divine genius and the perspicuity of Euclid'." 1 549. Joachim Camerarius published the enunciations of the first six Books in Greek and Latin (Leipzig). The book had a preface by Rhaeticus, a pupil of Copernicus, bom at Feldkirch in the Vorarlberg 1514, died 1576. Another edition with proofs of the propositions of the first three Books was published by Moritz Steinmetz in 1577 (Leipzig). 1550. loan. Scheubel published at Basel (also per loan. Her- vagiunt) the first six Books in Greek and Latin " together with true and appropriate proofs of the propositions, without the use of letters " (to denote points in the figures, the various straight lines and angles being described in words*). 1557 (also 1558). Stephanus Gracilis published another edition (repeated 1 573, 1 578, 1 598) of the enunciations (alone) of Books I. — xv. * Kiistner, I. p. a6o. ' Heiberg, vol. v. p. cvii. * K&stner, i. p. 359. % ...... INTRODUCTION i W^[ch. viii in Grcetf^tid Latin at Paris (afiud GuUelmum Caviilat\ He remarks tn thtf'preTace that for want of time he had changed scarcely anything /in 5doJcs L — VI., but m the remaining Books he had emended what [seemed obscure or inelegant in the Latin translation, while he had ' ad6pCed in its entirety the translation of Book X. by Pierre Mondor<i {^P^lrus Montaureus), published separately at Paris in 1551- Gracilis also added a few ** scholia." **; 1564, In this year Conrad Dasypodius (Rauchfuss), the inventor and maker of the clock in Strassburg cathedral, similar to the present one, which did duty from 1571 to 1789, edited (Strassburg, Chn Mylius) (i) Book L of the Elements in Greek and Latin with scholia, (2) Book IL in Greek and Latin with Barlaain*s arithmetical version of Book IL, and (3) the muncmticns of the remaining Books HI. — XllL Book L was reissued with "vocabula quaedam geometrica" of Heron, the enunciations of all the Books of the Eiemenis, and the other works of Euclid, all in Greek and Latin. In the preface to (l) he says that it had been for twenty-six years the rule of his school that all who were promoted from the classes to public lectures should learn the first Book, and that he brought it out, because there were then no longer any copies to be had, and in order to prevent a good and fruitful regulation of his school from falling through. In the preface to the edition of 1571 he says that the first Book was generally taught in all gymnasia and that it was prescribed in his school for the first class. In the preface to (3) he tells us that he published the enunciations of Books IIL — XIII, in order not to leave his work unfinished, but that, as it would be irksome to carry about the whole work of Euclid in extenso, he thought it would be more convenient to students of geometry to learn the Eiefnmts if they were compressed into a smaller book. t62a Henry Briggs (of Briggs* logarithms) published the first six Books in Greek with a Latin translation after Command in us, "corrected in many places" (London, G. Jones). 1703 is the date of the Oxford edition by David Gregory which, until the issue of Heiberg and Menge's edition, was still the only edition of the complete works of Euclid ^ In the Latin translation attached to the Greek text Gregory says that he followed Comman- dinus in the main, but correct^ numberless passages in it by means of the books in the Bodleian Library which belonged to Edward Bernard (1638- 1696), formerly Savilian Professor of Astronomy, who had conceived the plan of publishing the complete works of the ancient mathematicians in fourteen volumes, of which the first was to contain Euclid's Elements L — KV» As regards the Greek text, Gregory tells us that he consulted, as far as was necessary, not a few MSS. of the better sort, bequeathed by the great Savile to the University, as well as the corrections made by Savile in his own hand in the margin of the Basel edition. He had the help of John Hudson, Bodley s Librarian, who ^ ETEABtAOT tA ZCIZOMBKA. EucHdis q^e sapersunt omnut. Ex re<^nnotie D«TidU Grcgorii M.D, Astronomiae Professoris SavilUnt ci R.S,S, Oxoniac, c Tbeairo Sheldoniuto, An. Dom. iiDcciti. I CH. viii] TRANSLATIONS AND EDITIONS 103 punctuated the Basel text before it went to the printer, compared the Latin version with the Greek throughout, especially in the Elements and Data, and, where they differed or where he suspected the Greek text, consulted the Greek MSS. and put their readings in the margin if they agreed with the Latin and, if they did not agree, affixed an asterisk in order that Gregory might judge which reading was geo- metrically preferable. Hence it is clear that no Greek MS., but the Basel edition, was the foundation of Gregory's text, and that Greek MSS. were only referred to in the special passages to which Hudson called attention. 1 8 14-18 1 8. A most important step towards a good Greek text was taken by F. Peyrard, who published at Paris, between these years, in three volumes, the Elements and Data in Greek, Latin and French ^ At the time (1808) when Napoleon was having valuable MSS. selected from Italian libraries and sent to Paris, Peyrard managed to get two ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 204 was also at Paris at the time, but all three were restored to their owners in 18 14). Peyrard noticed the excellence of Cod. Vat. 190, adopted many of its readings, and gave in an appendix a conspectus of these readings and those of Gregory's edition ; he also noted here and there readings from Vat. .1038 and various Paris MSS. He there- fore pointed the way towards a better text, but committed the error of correcting the Basel text instead of rejecting it altogether and starting afresh. 1 824- 1 825. A most valuable edition of Books I. — ^vi. is that of J. G. Camerer (and C. F. Hauber) in two volumes published at Berlin". The Greek text is based on Peyrard, although the Basel and Oxford editions were also used. There is a Latin translation and a collection of notes far more complete than any other I have seen and well nigh inexhaustible. There is no editor or commentator of any mark who is not quoted from ; to show the variety of important authorities drawn upon by Camerer, I need only mention the following names : Proclus, Pappus, Tartaglia, Commandinus, Clavius, Peletier, Barrow, Borelli, Wallis, Tacquet, Austin, Simson, Playfair. No words of praise would be too warm for this veritable encyclopaedia of information. 1825. J. G. C. Neide edited, from Peyrard, the text of Books I. — VI., XI. and XII. {Halts Saxoniae). 1826-9. The last edition of the Greek text before Heiberg's is that of E. F. August, who followed the Vatican MS. more closely than Peyrard did, and consulted at all events the Viennese MS. Gr. 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains Books I. — XIII. ^ Etulidis quae iupersunt, Les (Euvres cTEuclide^ en Grec, en Laiin et en Fran^ais d^aprls un manuserit tris-ancien, qui itait restiinconnu jusqu^h nos jours. Par F. Peyrard. Ouvrage approuv^ par TlDStitut de France (Paris, chez M. Patris). * Euclidis elementorum Ubri sex pricres graece et laiine commentarU e scriptis veterum ac ncentiorum mathematuorum et PJUuieren maxime i/Zf/j/m/f (Berolini, sumptibus G. Reimeri). Tom. I. 1834; torn. n. 1835. I «4 INTRODUCTION r [ciu vui III, Latin versions or commentaries after 153?*""" 1545, Petrus Ramus (Pierre de la Ratnte* 15 15-1 572) is credited with a translation of Euclid which appeared in 1545 and again in 1 549 at Parish Ramus, who was more rhetorician and logician thanj geometer, also published in his ScAolae ma fA^malicae {i ^$g,Fr^nk{uTtl\ 1569, Basel) what amounts to a series of lectures on Euclid's Ekmtnts^^ in which he criticises Euclid's arrangement of his propositions, the; definittons, postulates and axioms^ all from the point of view of logiaj 1557. Demonstrations to the geometrical Elements of Euclid, six' Books, by Peletarius (Jacques Peleticr), The second edition (1610) contained the same with the addition of the "Greek text of Euclid**; but only the enunciatmis of the propositions, as well as the defini- tions etc*j are given in Greek (with a Latin translation), the rest is in Latin only* He has some acute observations, for instance about the "angle" of contact \ 1S59» Johannes Buteo, or Borrel (1492-1572), published in anj appendix to his book De quadraiura circuli some notes *'on the errors^ of Campanus, Zambertus, Orontius, Peletarius^ Pena, interpreters of Euclid/* Buteo in these notes proved, by reasoned argument based on original authorities, that Euclid himself and not Theon was the author of the proofs of the propositions* 1566. Franciscus Flussates Candalla (Francois de Foix, Comte de Candale, 1502-1594) "restored" the fifteen Books, following, as he says, the terminology of Zamberti s translation from the Greek, but drawing, for his proofs, on both Campanus and Theon (i,e, Zamberti) except where mistakes in tliem made emendation necessary. Other editions followed in 1578, 1602, 1695 (in Dutch). 1572. The most important Latin translation is that of Com- mandinus (i 509-1 575) of Urbino, since it was the foundation of most translations which followed it up to the time of Peyrard» including that of Simson and therefore of those editions, numerous in England, which give Euclid "chiefly after the text of Simson." Simson's first (Latin) edition (1756) has ''ex versione Latina Federici Commandini" on the title-page. Commandinus not only followed the original Greek more closely than his predecessors but added to his translation some ancient scholia as well as good notes of his own. The title of his work is Euclidis elementorum libri XV, una cum scholiis antiquis. A Federico Comtnandino Urbinate nuper in latinum conversi, commentariisque quibusdam iUustrati (Pisauri, apud CamiUum Francischinum). I He remarks in his preface that Orontius Finaeus had only edited ' six Books without reference to any Greek MS., that Peletarius had 1 followed Campanus' version from the Arabic rather than the Greek 1 text, and that Candalla had diverged too far from Euclid, having rejected as inelegant the proofs given in the Greek text and 1 substituted faulty proofs of his own. Commandinus appears to have 1 Described by Boiicom|Migni, BulkUmo^ II. p. 389. I CH. viii] TRANSLATIONS AND EDITIONS 105 used, in addition to the Basel editio princeps^ some Greek MS., so far not identified ; he also extracted his *' scholia antiqua " from a MS. of the class of Vat. 192 containing the scholia distinguished by Heiberg as "Schol. Vat" New editions of Commandinus' translation followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and Stone), 1756 (Books I. — vi., xi., xil. in Latin and English, by Simson), 1763 (Keill). Besides these there were many editions of parts of the whole work, e.g. the first six Books. 1574. The first edition of the Latin version by Clavius* (Christoph Schlussel, born at Bamberg 1537, died 161 2) appeared in 1574, and new editions of it in 1589, 1591, 1603. 1607, 1612. It is not a translation, as Clavius himself states in the preface, but it contains a vast amount of notes collected from previous commentators and editors, as well as some good criticisms and elucidations of his own. Among other things, Clavius finally disposed of the error by which Euclid had been identified with Euclid of Megara. He speaks of the diflferences between Campanus who followed the Arabic tradition and the " commentaries 01 Theon," by which he appears to mean the Euclidean proofs as handed down by Theon ; he complains of predecessors who have either only given the first six Books, or have rejected the ancient proofs and substituted worse proofs of their own, but makes an exception as regards Commandinus, ''a geometer not of the common sort, who has lately restored Euclid, in a Latin translation, to his original brilliancy." Clavius, as already stated, did not give a translation of the Elements but rewrote the proofs, com- pressing them or adding to them, where he thought that he could make them clearer. Altogether his book is a most useful work. 1 62 1. Henry Savile's lectures {Praelectiones tresdecim in prin- cipium Elenuntorum Euclidis Oxoniae habitae MDC.XX., Oxonii 162 1), though they do not extend beyond I. 8, are valuable because they grapple with the difficulties connected with the preliminary matter, the definitions etc., and the tacit assumptions contained in the first propositions. 1654. Andr6 Tacquet's Elementa geametriae planae et solidae containing apparently the eight geometrical Books arranged for general use in schools. It came out in a large number of editions up to the end of the eighteenth century. 1655. Barrow's Euclidis Elementorum Libri XV breviter demon-- strati is a book of the same kind. In the preface (to the edition of 1659) he says that he would not have written it but for the fact that Tacquet gave only eight Books of Euclid. He compressed the work into a very small compass (less than 400 small pages, in the edition of 1659, for the whole of the fifteen Books and the Data) by abbre- viating the proofs and using a large quantity of symbols (which, he says, are generally Oughtr^'s). There were several editions up to 1732 (those of 1660 and 1732 and one or two others are in English). 1 Euclidis eUmmtot-um lUni XV. Accessit xvi. de solidorum reguiarium comparatione, Omtus terspicuis demonstraHambuSt accuratisque sektdiis Ulustrati, Aucicre Ckristophoro Clamo (Romae, apud Vincentiam Accoltum), 1 vols. io6 INTRODUCTION [ch. viii 1658. Giacomo Alfonso Borelli (1608-1679) published EmMks resHtutus^ on apparently similar lines, which went Ummgh time mofe editions (one in Italian, 1663). 1660. Claude Franfois Milliet Dechales' eight geometrical Books of Euclid's Elements made easy. Dechales' versions of the Etemmts had great vogue, appearing in French, Italian and English as wdl as Latin. Riccardi enumerates over twenty editions. 1733. Sacdieri's EucUdes ab om$$i maeva vimUcatms swe emuOus geometricus quo stabiliuntur prima ^sa gmmutrieie frimci^ is important for his elaborate attempt to prove the parallel-postulate forming an important stage in the history of the demopment of non- Euclidean geometry. i7S6- Simson's first edition, in Latin and in English. The Latin title is Euclidis elementorum libri prions Mr, Uim mmd eeim us ei du0- decimus, ex versiotie latina Pederid Cammandini; sublaiis Us quibus olim Ubri hi a Tkeone^ aUisve, vitiaH sumi, ei quiiusdam Euclidis demonstrationibus resHiuiis. A Roberto SAmsom Jf.D. Glasguae, in aedibus Academids excuddMUit Robertus et Andreas Foulis, Academiae typografdiL 1802. Euclidis elementorum Ubri prims xn ex Com$nandimi ei Gregorii versionibus latinis. In usmn iuventuOs Aeademicae...hy Samuel Horsley, Bishop of Rochester. (Oxft>rd, Clarendcm Press.) IV. Italian versioms or oomcentaries. 1543. Tartaglia's version, a second edition of which was pub- lished in 1565^ and a third in 1585. It does not appear that he used anv Greek text, for in the edition of is6s he mentions as available only ''the first translation by Campano," "the second made by Bartolomeo Zamberto Veneto who is still alive," "the editions of Paris or Germany in which they have included both the aforesaid translations," and "our own translation into the vulgar (tongue)." 1575. Commandinus' translation turned into Italian and revised by him. 161 3. The first six Books "reduced to practice" by Pietro Antonio Cataldi, re-issued in 1620, and followed by Books vii. — ix. (1621) and Book x. (1625). 1633. Borelli's Latin translation turned into Italian by Domenico Magni. 1680. Euclide restituto by Vitale Giordano. 169a Vincenzo Viviani's Elementi piani e solidi di Eudide (Book v. in 1674). 1 The title-ptge of the edition of 1565 is as follows : EuciuU Magurrmu phiUsoptU^ s§U imirmbitUrt dMe seimiie imaiMsmtOieit diligmUmsnte tyusifUU^, ei alia inttgritH ridotu^ per U degmo prefessore di tal seientie NicoU Teartaka Briseianp, stimdo U due traJottiam, eon una empla etpositwme dtlU istaso traddtore di m$iwc aggiwUa, talmenU chiara^ ctu ogm wudiccre ingegm*^ stmna la MOtilia, otter sujfragio di oUtnT altra uimtia emt facUiU jm empau a poiirU imUmdtn^ In Veneda, Appresio Cmtio Troiano, 1565. ^ CH. viii] TRANSLATIONS AND EDITIONS 107 173 1. EUnufiti geametrici piani e solidi di Euclide by Guido Grandi. No translation, but an abbreviated version, of which new editions followed one another up to 1806. 1749. Italian translation of Dechales with Ozanam*s corrections and additions, re-issued 1785, 1797. 1752. Leonardo Ximenes (the first six Books). Fifth edition, 1819. 18 1 8. Vincenzo Flauti's Corso di geotnetria eUmentare e sublime (4 vols.) contains (Vol. I.) the first six Books, with additions and a dissertation on Postulate 5, and (Vol. II.) Books XL, XII. Flauti also published the first six Books in 1827 and the Elements of geometry of Euclid in 1843 and 1854. V. German. 1558. The arithmetical Books vil.— ix. by Scheubel* (cf. the edition of the first six Books, with enunciations in Greek and Latin, mentioned above, under date 1550). 1562. The version of the first six Books by Wilhelm Holtzmann (Xylander)*. This work has its interest as the first edition in German, but otherwise it is not of importance. Xylander tells us that it was written for practical people such as artists, goldsmiths, builders etc., and that, as the simple amateur is of course content to know facts, without knowing how to prove them, he has often left out the proofs altogether. He has indeed taken the greatest possible liberties with Euclid, and has not grappled with any of the theoretical difficulties, such as that of the theory of parallels. 165 1. Heinrich Hoffmann's Teutscher Euclides {2nd edition 1653), not a translation. 1694. Ant Ernst Burkh. v. Pirckenstein's Teutsch Redender Euclides (eight geometrical Books), "for generals, engineers etc." "proved in a new and quite easy manner." Other editions 1699, 1744. 1697. Samuel Reyher's In teutscher Sprache vorgestellter Euclides (six Books), "made easy, with symbols algebraical or derived from the newest art of solution." 1 7 14. Euclidis XV Backer teutsch, "treated in a special and brief manner, yet completely," by Chr. Schessler (another edition in 1729). 1773. The first six Books translated from the Greek for the use of schools by J. F. Lorenz. The first attempt to reproduce Euclid in German word for word. 1 78 1. Books XL, XII. by Lorenz (supplementary to the pre- ceding). Also EuklicTs Elemente funfzehn Bucher translated from ^ Das sibendacht und nmnt buck des hochberiimbten Mathematici Euclidis Megarensis... durch Magistrum Johann Sckeybl^ der loblichen univtrsiUt tu Tiibingaty des Euclidis und Arithmclic Ordinaritn^ auss dem latein ins teutsch gebracht..., • Die seeks erste Biicher Euclidis vom an fang oder grundder Geometry... Auss Griechiscker sprach in die Teiitsck gebracht aigentlich erkldrt...Demcusen vormals in TeUtseher sprach nie gueken warden... Durch Wilhelm HoUtman genani Xylander von Augspterg. Getruckht <a io8 INTRODUCTION [cH. vui the Greek by Lorenz (second edition 1798 ; editions of 1809^ 1818, 1824 by MoUweide, of 1840 by Dippe). The edition of 1834, and I presume those before it, are shortened by the use of symbols and the compression of the enunciation and ^setting-out" into one. 1807. Books I.— VL, Xl^ XIL ''newly transuited from the Greekp" by J. K. F. HaufT. 1828. The same Books by Joh. Jos. Ign. Hoffmann ''as guide to instruction in elementary geometry, followed in 1832 by obmrva- tions on the text by the same editor. 1833. Die GeometrU des BukUd und das Wksm dirsMm by E. S. Unger; also 1838, 185 1. 1901. Max Simon, Ettclid und dU Sichs planimetrischm Bikfur. VI. French. 1564-1566. Nine Books translated by Pierre Forcadd, a pupil and friend of P. de la Ramde. 1604. The first nine Books translated and annotated by Jean Errard de Bar-le-Duc; second edition, 1605. 161 5. Denis Henrion's translation of the 15 Books (seven editions up to 1676). 1639. The first six Books ^demonstrated by symtxds, by a method very brief and intelligible^'' by Pierre Hdngone, mentioned by Barrow as the only editor who, before him, had lued symbols for the exposition of Euclid. 1672. Eight Books "rendus plus fiidles*' by Claude Francois Milliet Dechales, who also brought out Les iUtnens d'Euclide ex- pliquis d*un€ maniire nouvelle et trh facile^ which appeared in many . editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), I and was translated into Italian (1749 etc.) and English (by William ^ Halifax, 1685). 1 1804. In this year, and therefore before his edition of the Greek text, F. Peyrard published the Elements literally translated into French. A second edition appeared in 1809 ^^^ ^^ addition of the fifth Book. As this second edition contains Books I. — vi. XI., XII. and X. I, it would appear that the first edition contained Books I. — ^iv., VI., XL, XII. Peyrard used for this translation the Oxford Greek text and Simson. VII. Dutch. 1606. Jan Pieterszoon Dou (six Books). There were many later editions. Kastner, in mentioning one of 1702, says that Dou explains in his preface that he used Xylander's translation, but, having after- wards obtained the French translation of the six Books by Errard de Bar-le-Duc (see above), the proofs in which sometimes pleased him more than those of the German edition, he made his Dutch version by the help of both. 16 1 7. Frans van Schooten, "The Propositions of the Books of Euclid's Elements'*; the fifteen Books in this version ''enlarged" by Jakob van Leest in 1662. 1695. C. J. Voc^ht, fifteen Books complete, with Candalla's •' 16th.'' CH. viiij TRANSLATIONS AND EDITIONS 109 1702. Hendrik Coets, six Books (also in Latin, 1692); several editions up to 1752. Apparently not a translation, but an edition for school use. 1763. Pybo Steenstra, Books I. — Vl., XL, XII., likewise an abbre- viated version, several times reissued until 1825. 1 VIII. English. f 1570 saw the first and the most important translation, that of Sir Henry Billingsley. The title-page is as follows : THE ELEMENTS OF GEOMETRIE of the most auncient Philosopher EVCLIDE of Megara Faithfully {now first) translated into the Englishe toung^ by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies^ Annotations^ and Inuentions^ of the best Mathematidens, both of time pasty and in this our age. With a very fruitfull Preface by M. I. Dee, specifying the chiefe Mathematicall Scieces, what they are^ and whereunto commodious: where^ also, are disclosed certaine new Secrets Mathematicall and Mechanically vntill these our daies, greatly missed. Imprinted at London hy fohn Daye. The Preface by the translator, after a sentence observing that with- out the diligent study of Euclides Elementes it is impossible to attain unto the perfect knowledge of Geometry, proceeds thus. " Wherefore considering the want and lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to their countrey in those of our nation, to whom God hath geuen both knowledge and also abilitie to translate into our tounge, and to Ipublishe abroad such good authors and bookes (the chiefe instrumentes of all leaminges): seing moreouer that many good wittes both of gentlemen and of others of all degrees, much desirous and studious of F these artes, and seeking for them as much as they can, sparing no paines, and yet frustrate of their intent, by no meanes attaining to that which they seeke : I haue for their sakes, with some charge and great trauaile, faithfully translated into our vulgare touge, and set abroad in Print, this booke of Euclide. Whereunto I haue added easie and plaine declarations and examples by figures, of the defini- tions. In which booke also ye shall in due place finde manifolde I additions, Scholies, Annotations, and Inuentions: which I haue gathered out of many of the most famous and chiefe Mathematicies, both of old time, and in our age : as by diligent reading it in course, ye shall well perceaue...." It is truly a monumental work, consisting of 464 leaves, and there- fore 928 pages, of folio size, excluding the lengthy preface by Dee. The notes certainly include all the most important that had ever been no INTRODUCTION [ck. vui written, from those of the Greek commeiitators, Proclus and the others whom he quotes, down to those of Dee himself on the last books. Besides the fifteen Books, Billingsley included the ''sixteenth" added by Candalla. The print and appearance of the book are worthy of its contents ; and, in order that it may be understood how no pains were spared to represent everything in the dearest and most perfect form, . I need only mention that the figures of the propositions in Book XL J are nearly all duplicated, one being the figure of Euclid, the other an 1 arrangement of pieces of paper (triangular, rectang^ular etc) pasted at the edges on to the page of the book so that the pieces can be turned up and made to show the real form of the solid figures represented. Billingsley was admitted Lady Margaret Scholar of St John's College, Cambridge, in 1 55 1, and he is also said to have studied at i Oxford, but he did not take a d^ree at either University. He was ^ afterwards apprenticed to a London haberdasher and rapidly became a wealthy merchant Sheriff of London in 1584, he was elected Lord Mayor on 3 ist December, 1 596^ on the death, during his year of oflSce^ of Sir Thomas Skinner. From 1589 he was one of the Queen's four " customers,'* or farmers of customs, of the port of London. In 1591 he founded three scholarships at St John's Collie for poor students, and gave to the College for thdr noAintenance two messuages and tenements in Tower Street and in Mark Lane, AllhallowSi Barking. He died in 1606. 165 1. Elements of Geometry, The first VI Boocks: In a eompeti^ diousform contracted and demomtraUd by Captain Thomas Rudd, with the mathematicall preface of John Dee (London). 1660. The first English edition of Barrow's Euclid (published in Latin in 1655), appeared in London. It contained "the whole fifteen books compendiously demonstrated"; several editions followed, in 1705, 1722, 1732, 1751. 1661. Eudids Elements of Geometry^ with a supplement of divers Propositions and Corollaries, To which is added a Treatise of regular Solids by Campane and Flussat ; likewise EuclicTs Data and Marinus his Preface. Also a Treatise of the Divisions of Superficies^ ascribed to Machomet Bagdedine^ but published by Commandine at the request of f. Dee of London. Published by care and industry of John Leeke and Geo. Serle, students in the Math. (London). According to Potts this was a second edition of Billingsley's translation. 1685. William Halifax's version of Dechales' " Elements of Euclid i explained in a new but most easy method " (London and Oxford). 1705. The English Euclide; being the first six Elements of I Geometry^ translated out of the Greeks with annotations and useful/ ( supplements by Edmund Scarburgh (Oxford). A noteworthy and 4 useful edition. \ 1708. Books L — ^VL, XL, XII., translated from Commandinus' Latin version by Dr John Keill, Savilian Professor of Astronomy at Oxford. Keill complains in his preface of the omissions by such editors as Tacquet and Dechales of many necessary propositions (e.g. VL 27 — 29), and of their substitution of proofs of their own for Euclid's. He praises Barrow's version on the whole, though objecting to the ** algebraical " CH. viii] TRANSLATIONS AND EDITIONS iix form of proof adopted in Book ll., and to the excessive use of notes and symbols, which (he considers) make the proofs too short and thereby obscure: his edition was therefore intended to hit a proper mean between Barrow's excessive brevity and Clavius' prolixity. Keiirs translation was revised by Samuel Cunn and several times reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782 the twelfth. 1 7 14. W. Whiston's English version (abridged) of The Elements of Euclid with select theorems out of Archimedes by the learned Andr. Tacguet. 1756. Simson's first English edition appeared in the same year as his Latin version under the title : The Elements of Euclid, viz, the first six Books together with the eleventh and twelfth. In this Edition the Errors by which Theon or others have long ago vitiated these Books are corrected mtd some of Euclid's Demonstrations are restored. By Robert Simson (Glasgow). As above stated, the Latin edition, by its title, purports to be "ex versione latina Federici Commandini,*' but to the Latin edition, as well as to the English editions, are appended Notes Critical and Geometrical; containing an Account of those things in which this Edition differs from the Greek text; and ike Reasons of the Alterations which have been made. As also Obser- vations on some of the Propositions. Simson says in the Preface to some editions (e.g. the tenth, of 1799) ^^^^ '*^^ translation is much amended by the friendly assistance of a learned gentleman." Simson's version and his notes are so well known as not to need any further description. The book went through some thirty suc- cessive editions. The first five appear to have been dated I7S6, 1762, 1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806, the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844. The DcUa '' in like manner corrected " was added for the first time in the edition of 1762 (the first octavo edition). 1 78 1, 1788. In these years respectively appeared the two volumes containing the complete translation of the whole thirteen Books by James Williamson, the last English translation which reproduced Euclid word for word. The title is The Elements of Euclid, with Dissertations intended to assist and encourage a critical examination of these Elements, as the most effectual means of establishing a juster taste upon mathematical } subjects than that which at present prevails. By James Williamson. tin tfie first volume (Oxford, 1781) he is described as "MA. Fellow of Hertford College," and in the second (London, printed by T. Spilsbury, 1788) as "B.D." simply. Books v., vi. with the Con- ) elusion in the first volume are paged separately from the rest 1 78 1. An examination of the first six Books of Euclid's Elements, by William Austin (London). 1795- John Playfair's first edition, containing "the first six Books of Euclid with two Books on the Geometry of Solids." The book INTRODUCTION [CB. Tin a fifth edition in 1819, an eighth in i83l,a ninth in 1836^ and Ith in 1846. [826. Riccardi notes under this* date Euclid's EUmmis 9f Gta- * containing the whole twelve Books translmtidinio Et^gKsktfr^m ike ofPeyrard^ by George Phillips. The editor, who waft rasident FQueens' College, Cambridge, 1857-1892, was bom in 1804 and jltriculated at Queens' in 1826, so that he must have published the 3k as an undergraduate. 1828. A very valuable edition of the fiist six Books is diat of l^ionysius Lardner, with commentary and ge om etrical exerdses, to Irhich he added, in place of Books XL, XIL, a Treatise on Solid reometry mostly based on L^endre. Lardner compresses the pro- sitions by combining the enunciation and tihe setting-out, auid lie [gives a vast number of riders and additional propositions in smaller print The book had reached a ninth edition by 184^ and an eleventh by 185s. Among other things, Lardner gives an Appendix ^on the theory of parallel lines," in which he gives a short histoiy of the attempts to get over the difficulty of the parallel-postulate^ down to that of Legendre. 1833. T. Perronet Thompson's Gmmutvy wiUumt axioms^ or ike first Book of EuclicTs Elements wiik aUerwOons emd fMes; emd em intercalary book in which the straight Urn emd fleme are dnkfod from properties of tlu sphere ^ with an appendix coniamit%g noOces of m e ikods proposed for getting over the diffictUty in ike iwe^k euHam ofEneUd Thompson (1783-1869) was 7th wrangler 1802, mklshipman 1803, Fellow of Queens' College, Cambridge, 1804, and afterwards general and politician. The book went through several editions, but» naving been well translated into French by Van Tenac» is nid to have received more recognition in France than at home. 1845. Robert Potts' first edition <(and one of the best) entitled: Euclid's Elements of Geometry chiefly from the text of Dr Simson with explanatory notes.. Jo which is prefixed an introduction containing a brief outline of the History of Geometry. Designed for tlu use of the higher forms in Public Schools a»id students in the Universities (Cambridge University Press, and London, John W. Parker), to which was added (1847) ^^ Appendix to the larger edition of Euclid s Elements of Geometry^ containing additional notes on the Elements, a short tract on trans- versalSy and hints for the solution of the problems etc. 1862. Todhunter's edition. The later English editions I will not attempt to enumerate ; their name is l^ion and their object mostiy that of adapting Euclid for school use, with all possible gradations of departure from his text and order. IX. Spanish. 1576. The first six Books translated into Spanish by Rodrigo ^amorano. 1637. The first six Books translated, with notes, by L. Carduchi. 1689. Books L — VL, XL, xiL, translated and explained by Jacob Knesa. 1 CH. viii] TRANSLATIONS AND EDITIONS 113 X. Russian. 1739. Ivan Astaroff (translation from Latin). 1789. Pr. SuvorofTand Yos. Nikitin (translation from Greek). 1 880. Vachtchenko-Zakhartchenko. (18 1 7. A translation into Polish by Jo. Czecha.) XI. Swedish. 1744. Mirten Stromer, the first six Books; second edition 1748. The third edition (1753) contained Books XI. — xii. as well; new editions continued to appear till 1884. 1836. H. Falk, the first six Books. 1844, J 84s, 1859. P. R. BrSkenhjelm, Books I. — VI., XI., XII. i8sa F. A. A. Lundgren. 1850. H. A. Witt and M. E. Areskong, Books I.— vi., xi., xil. XII. Danish. 174s. Ernest Gottlieb Ziegenbalg. 1803. H. C. Linderup, Books I. — VI. XIII. Modern Greek. 1820. Benjamin of Lesbos. H. E. -*■ -i^- -t.V* ,Hr > CHAPTER IX. { 1, ON THE NATURE OF ELEMENTS. It would not: be easy to find a more lucid explanation of the terms tknunt and eUmentafy, and of the distinction between them, than is found in Proc1us\ who is doubtless, here as so often, quoting from Geminus, There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of I a principle, all- pervading, and furnishing proofs of many properties* Such theorems are called by the name of tlemintsx and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek The term elententaty, on the other hand, has a wider application : it is applicable to things "which extend to greater multiplicity, and, though possessing simplicity and elegance, have no longer the same dignity as the eUments^ because their investigation is not of general use in the whole of the science, eg, the proposition that in triangles the perpendiculars from the angles to the trans vei^e sides meet in a point." " Again, the term element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one anotiier. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)*, and via versa. Such an element is like a lemma. But the term element is otherwise used of that into which, being more simple, the composite is divided ; and in this sense we can no longer say that everything is an element of everything, but only that things which are more of the ^ nature of principles are elements of those which stand to them In the relation of results, as postulates are elements of theorems. It is ^ Prodos, Comm, on Etui, i., ed. Friedleio, f>p. 7a sqq. * T^ vX^of rfir hrrht 6p0tut f^wr. If the text is nght, we most apparently take it as ''the number of the angles equal to right angles that there are inside,'* i.e. that are made up by the internal angles. I CH. IX.J1] ON THE NATURE OF ELEMENTS 115 according to this signification of the term element that the elements found in Euclid were compiled, being partly those of plane geometry, and partly those of stereometry. In like manner many writers have drawn up elementary treatises in arithmetic and astronomy. *' Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest proceeds, and into which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less ; some used shorter proofs, some extended their investigation to an indefinite length ; some avoided the method of reductio ad absurdum^ some 2l\o\A^ proportion \ some contrived preliminary steps directed against those who reject the principles ; and, in a word, many different methods have been invented by various writers of elements. " It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge) ; it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science) ; it must have great r^ard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest ; for its utility avails towards the investigation of the primordial figures*, its clearness and organic perfection are secured by the precession from the more simple to the more complex and by the foundation of the investigation upon common notions, while generality of demonstration is secured by die progression through the theorem's which are primary and of the nature of principles to the things sought. ! As for the things which seem to be wanting, they are partly to be 1 discovered by the same methods, like the construction of the scalene • and isosceles (triangle), partly alien to the character of a selection of elements as introducing hopeless and boundless complexity, like the ! subject of unordered irrationals which Apolionius worked out at length', and partly developed from things handed down (in the elements) as causes, like the many species of angles and of lines. These things then have been omitted in Euclid, though they have received full discussion in other works ; but the knowledge of them is derived from the simple (elements)." Proclus, speaking apparently on his own behalf, in another place distinguishes two objects aimed at in Euclid's Elements. ZTEe first has reference to the matter of the investigation, and here, li^e a good Platonist, he takes the whole subject of geometry to be concerned with the " cosmic figures," the five regular solids, which in Book XIIL * r(a9 d^x^'f^ ^iiM^rciir, by which Proclus probably means the regular polyhedxa (Tannery, p. i43»-)- ' We have no more than the most obscure indications of the character of this work in an . Arabic MS. analysed by Woepcke, Essed d^une restitution de travaux perdm dAtolUmius tur Us fuantith irrationelUs d*apris da indicaticHS tirits d*un manuscrit arabi in Mhnoires fristntis h ttuadhnie da sciences^ XI v. 658—720, Paris, 1856. Cf. Cantor, Gtsch. d, Maik. 't* PP* 34^~~9 * details are also given in my notes to Book x. i' 8-a li 1 16 INTRODUCTION [CH. dl f i are constructed, inscribed in a sphere and compared with one another. The second object is relative to the learner; and, from this standpoint, ^the elements may be described as ''a means of perfecting the learnei's understanding with reference to the whole of geometiy. | For, starting from these (elements), we shall be able to acquire knowledge of the other parts of this science as well, while without them it is impossible for us to get a grasp of so complex a subject, and knowledge ci the rest is unattainable. As it is, the theorems which are most of the nature of principles, most simple, and most akin to the first hypotheses are here collected, in their appropriate order; and the proon of all other propositions use these theorems as thoroughly well known, and start from them. Thus Archimedes in the books on the sphere and cylinder, Apollonius, and all other geometers, clearly use the theorems proved in this very treatise as constituting admitted principles\'' Aristotle too speaks of elements of geometry in the same sense. Thus: "in geometry it is well to be thoroughly versed in the elements*"; "in general the first of the elements are, given the definitions, e.g. of a straight line and of a circle^ most easy to prove, although of course there are not many data that can be used to establish each of them because there are not many middle terms*"; "among geometrical propositions we call those 'elements' the proofs of which are contained in the proofs of all or most of such propositions^*'; "(as in the case of bodies)^ so in like manner we speak of the elements ' of geometrical propositions and, generally, of demonstrations ; for the demonstrations which come first and are attained in a variety of other demonstrations are called elements of those demonstrations... the term element is applied by analogy to that which, being one and small, is useful for many purposes'." § 2. ELEMENTS ANTERIOR TO EUCLID'S. The early part of the famous summary of Proclus was no doubt drawn, at least indirectly, from the history of geometry by Eudemus ; this is generally inferred from the remark, made just after the mention of Philippus of Mende, a disciple of Plato, that "those who have written histories bring the development of this science up to this point" We have therefore the best authority for the list of writers of elements given in the summary. Hippocrates of Chios (fl. in second half of 5th c.) is the first ; then Leon, who also discovered diarismiy put together a more careful collection, the propositions proved in it being more numerous as well as more serviceable*. Leon was a little older than Eudoxus (about 390-337 B.C) and a little younger than Plato (429-348 B.C.), but did not belong to the latter's school. The * Prodtts, pp. 70, 10 — 71, ai. * T^s VIII. 14, 163 b as. » To^s viii. 3, 158 b 35. * Mett^k. 998 a as. * ifaath, loi^a 35— b 5. * Prodos, p. 66, ao iSrrt rdr Atforra iral tA 9roix«<a «wtft4Mu r^ rt rX^^i k9X rj %pd^ rQf9 dmnnf/thttif iwifUkkm^. I CH. ix.§a] ELEMENTS ANTERIOR TO EUCLID'S 117 geometrical text-book of the Academy was written by Theudius of Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus consorted together in the Academy and carried on their investigations in common. Theudius " put together the elements admirably, making many partial (or limited) propositions more generals" Eudemus mentions no text-book after that of Theudius, only adding that Her- motimus of Colophon "discovered many of the elements"." Theudius then must be taken to be the immediate precursor of Euclid, and no doubt Euclid made full use of Theudius as well as of the discoveries of Hermotimus and all other available material. Naturally it is not in Euclid's Elements that we can find much light upon the state of the subject when he took it up ; but we have another source of informa- tion in Aristotle. Fortunately for the historian of mathematics, Aristotle was fond of mathematical illustrations ; he refers to a con- siderable number of geometrical propositions, definitions etc., in a way which shows that his pupils must have had at hand some text- book where they could find the things he mentions; and this text-book must have been that of Theudius. Heiberg has made a most valuable collection of mathematical extracts from Aristotle*, from which much is to be gathered as to the changes which Euclid made in the methods of his predecessors ; and these passages, as well as others not included in Heiberg's selection, will often be referred to in the sequel. §3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, AND AXIOMS. On no part of the subject does Aristotle give more valuable information than on that of the first principles as, doubtless, generally accepted at the time when he wrote. One long passage in the Posterior Analytics is particularly full and lucid, and is worth quoting in extenso. After laying it down that every demonstrative science starts from necessary principles^ he proceeds': ^ By first principles in each genus I mean those the truth of which It is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed ; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed) ; and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of Uie premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the sub- ject-matter included under the particular science. Instances of first ^ Proclus, P' 67, 14 icoi 7dp rd rroixeSa Ka\(at avpira^tp koI roXKii rOif fitpucwp [6piKQp (?) Friedlein] KaBoXucirtpa iwolyfatw. ' Proclus, p. 67, 33 Twv CTOixtif^ roXXd dptvp€. * Maikemaiiscka mu AristoUies in Abhandlungen wur Gtsch, d, math. Wissenichafttn^ xvni. Heft (1904), pp. I— 49* ^ Anal, post, I. 6, 74 b 5. * ibitL i. 10, 76 a 31 — 77 a 4. 1 18 INTRODUCTION [ch. ul f 3 principles peculiar to a science are fhe assumptions that a Une is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each tA the common principles is true so far as regards the particular genus (subject-matter) ; for (in geometiy) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers. '* Now the things peculiar to the science, the existence of which must be assumed, are the things with r efer en ce to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed : thus arithmetic assumes the answer to die question what is (meant by) 'odd* or 'even,' 'a square' or 'a cube,' and geometry to the question what is (meant by) 'the irrational ' or 'deflection' or (the so-called) 'verging' (to a point); but that there are such things is proved by means of the common principles and of what has already been demonstrated. Similarly with astronomy. For every demonstrative science has to do with three things, (1) tne things wUch are assumed to exist, namely the genus (subject-matter) in ei^ case, the essential properties of which the science investigates, (2) the common axioms so-called, which are the prinuuy source of demonstration, and (3) the properties with regard to which all that is assumed is the meaning of the respective terms used. There is, however, no reason why some sciences should not omit to speak of one or other of these things. Thus there need not be any supposition as to the existence of the genus, if it is manifest that it exists (for it is not equally clear that number exists and that cold and hot exist) ; and, with regard to the properties, there need be no assumption as to the meaning of terms if it is clear : just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equads from equals, because it is well known. But none the less is it true that there are three things naturally distinct, the subject-matter of the proof, the things provod, and the (axioms) from which (the proof starts). •'Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demon- stration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. New anything that the teacher assumes, though it is matter of proft, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not abso- lutely, but relatively to the particular pupil ; but, if the same thing b assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference CH. DL§3] FIRST PRINCIPLES 119 between a hypothesis and a postulate ; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or I whatever is assumed and used without being proved, although matter Ffor demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood : a definition is therefore not a hypothesis, unless indeed it be asserted f that any audible speech is a hypothesis. A hypothesis is that from I the truth of which, if assumed, a conclusion can be established. Nor i are the geometer's hypotheses false, as some have said : I mean those ^ who say that ' you should not make use of what is false, and yet the I geometer falsely calls the line which he has drawn a foot long when t it is not, or straight when it is not straight' The geometer bases no ' conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither" (because the subject is of equal extent with what is predicated of it). Every demonstrative science, says Aristotle, must start from in- demonstrable principles : otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are {b) particular, or peculiar to the particular science ; {a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. Coming now to (b) the principles peculiar to the particular science which must be assumed, we have first the genus or subject-matter, the existence of which must be assumed, viz. magni- tude in the case of geometry, the unit in the case of arithmetic. Under this we must assume definitions of manifestations or attributes of the genus, e.g. straight lines, triangles, deflection etc. The definition in itself says nothing as to the existence of the thing defined : it only requires to be understood. But in geometry, in addition to the genus and the definitions, we have to assume the existence of a kvf frimary things which are defined, viz. points and lines only : the existence of everything else, e.g. the various figures made up of these, as triangles, squares, tangents, and their properties, e.g. incommensur- ability etc., has to be proved (as it is proved by construction and demonstration). In arithmetic we assume the existence of the unit: but, as regards the rest, only the definitions, e.g. those of odd, even, square, cube, are assumed, and existence has to he proved. We have then clearly distinguished, among the indemonstrable principles, axioms and definitions, A postulate is also distinguished from a hypothesis, the latter being made with the assent of the learner, the former iRthout such assent or even in opposition to his opinion (though, strangely enough, immediately after saying this, Aristotle gives a wider meaning to "postulate" which would cover "hypothesis" as well, namely whatever is assumed, though it is matter for proof, and used without being proved). Heiberg remarks that there is no trace in Aristotle of Euclid's Postulates, and that " postulate" in Aristotle has X 20 INTRODUCTION [ch. xx. f 3 a different meaning. He seems to base this on the alternative description of postulate, indistinguishable from a hypothesis; but. if we take the other description in which it is distinguished fiom a hypothesis as being an assumption of something which is a proper subject of demonstration without the assent or against the opinion of the learner, it seems to fit Euclid's Postulates fainy well, not only the first three (postulating three constructionsX but eminently also the other two, that all right angles are equal, and that two straight lines meeting a third and making the internal angles on the same side of it less than two right angles will meet on that side. Aristotle's description also seems to me to suit the ''postulates" mth which Archimedes b^ins his book On the equilibrium ofplams, namely that equal weights baliuice at equal distances, and that equal weights at unequal distances do not balance but that the weight at the longer dbtance will prevail Aristotle's distinction also between kfpothisis and definiiiaH^ and between hypothesis and eua&m^ is clear from the foUowine passa^: ''Among immediate syllogistic principles, I call that a msis which it is neither possible to prove nor essential for any one to hold who is to learn anything ; but that which it is necessary for any one to hold who is to learn anything whatever is an axiom : for there are some principles of this kind, and that is the most usual name 1^ which we speak of them. But, of theses, one kind is that which assumes one or other side of a predication, as, for instance^ that something exists or does not exist, and this is a Ig^thisis ; the other, which makes no such assumption, is a defimiioH. For a definition is a thesis : thus the arithmetician posits (rlOrrai) that a unit is that which is indivisible in respect of quantity ; but this is not a hypo- thesis, since what is meant by a unit and the fact that a unit exists are different things*." Aristotle uses as an alternative term for axioms ''common (things)," ra tcoivd, or "common opinions" (tcoipol So^ai), as in the following passages. *' That, when equals are taken from equals, the remainders are equal is (a) common (principle) in the case of all quantities, but mathematics takes a separate department (diroXafiovaa) and directs its investigation to some portion of its proper subject-matter, as e.g. lines or angles, numbers, or any of the other quantities"." "The common (principles), e.g. that one of two contradictories must be true, that equals taken from equals etc, and the like*...." " Withr^[ard to the principles of demonstration, it is questionable whether they belong to one science or to several. By principles of demonstration I mean the common opinions from which all demonstration proceeds, e.g. that one of two contradictories must be true, and that it is impossible for the same thing to be and not be^" Similarly "every demonstrative (science) investigates, with r^ard to some subject-matter, the essential attributes, starting from the common opinions*** We have then here, as Heiberg says, a sufficient explanation of Euclid's term for axioms, * Anai, post. I. a, 7a a 14—94. * Mttaph, 1061 b 19—34. * Anai. past, 1. 11, 77 a 30. « Mtuipk. 996b 36—30. * Mdapk. 997 a so— ^3. CH. ix.§3] FIRST PRINCIPLES lai viz. common notions {koivoX hpouti), and there is no reason to suppose it to be a substitution for the original term due to the Stoics : cf. Proclus* remark that, according to Aristotle and the geometers, axiom and common notion are the same things Aristotle discusses the indemonstrable character of the axioms in the Metaphysics. Since "all the demonstrative sciences use the axiomsV' the question arises, to what science does their discussion belong'? The answer is that, like that of Being {ovala), it is the province of the (first) philosopher^ It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations : if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axioms* ; thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principles*. It is ignorance alone that could lead any one to try to prove the axioms' ; the supposed proof would be a petitio principii*. If it is admitted that not every- thing can be proved, no one can point to any principle more truly indemonstrable*. If any one thought he could prove them, he could at once be refuted ; if he did not attempt to say anything, it would be ridiculous to argue with him : he would be no better than a vegetable". The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demon- stration. And so on". It was necessary to give some sketch of Aristotle's view of the first principles, if only in connexion with Proclus' account, which is as follows. As in the case of other sciences, so ''the compiler of elements in geometry must give separately the principles of the science, and after that the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them : they are treated as self-evident... Thus the first essential was to dis- tinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into hypotheses^ postulates^ and axioms. For all these are different from one another : an axiom, a postulate and a hypothesis are not the same thing, as the inspired Aristotle somewhere says. But, whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom^ e.g. the statement that things which are equal to the same thing are also equal to one another. When, on ^ Proclus, p. 104, 8. * Mdaph. 907 a 10. ' ibid, 99(5 36. ^ ibid, 1005 a 31— b 11. * ibid. 997 a 5 — 8. * ilnd. 1005 b II — 17. ' ibid. 1006 a 5. * ibid. 1006 a 17. * fA«/. 1006a la ^ f!^f^ 1006a II— 15. " ftW. 1006 a 18 sqq. &. 123 INTRODUCTION [cB. n. f 3 the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless iBys it down and assents to its being assumed, such an assumption is a Ig^thnis, Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When again what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postidaU^ e.g. that all right angles are equal This view of a postulate is clearly implied by those who have made a spedal and systematic attempt to show, with r^[ard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate and a hypothesis are thus distinguished^** We observe, first, that Proclus in this passage confuses kypotlksis and definitions, although Aristotle had made the distinction quite plain. The confusion may be due to his having in his mind a passage of Plato from which he evidently got the phrase about ^ not giving an account of" the principles. The passage is*: ^ I think you know that those who treat of geometries and calculations (arithmetic) and such things take for granted (Airotf^/Myoi) odd and even, figures, angles of three kinds, and other things akin to these in each subject, implying that they know these things, and, though using them as hypotheses, do not even condescend to give any account of them either to themselves or to others, but b^n from these things and then go through everything else in order, arriving ultimatdy, l^ recognised methods, at the conclusion which they started in search of." But the hypothesis is here the assumption, e.g. * that there may be suck a thing as length without breadth, henceforward called a line*,' and so on, without any attempt to show that there is such a thing ; it is mentioned in connexion with the distinction between Plato's 'superior' and 'inferior' intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good. We pass now to Proclus' account of the difference between postu- lates and axioms. He begins with the view of Geminus, according to which " they differ from one another in the same way as theorems are also distinguished from problems. For, as in theorems we propose to see and determine what follows on the premisses, while in problems we are told to find and do something, in like manner in the axioms such things' are assumed as are manifest of themselves and easily apprehended by our untaught notions, while in the postulates we . assume such things as are easy to find and effect (our understanding suffering no strain in their assumption), and we require no complication of macluneryV'..."Both must have the characteristic of being simple * Proclns, pp. 75, 10—77, *• * RipubHc, VI. CIO c. CJf. Aristotle, Nic. Eth^ 1 151 a 17. * H. }%c\aKmtjaurHal of Phiiohgy, vol. X. p. 144. ^ Produs, pp. 178, 13—179, 8. In Ulnstntion Prodos contrasts the drawins of a straight line or a circle with the drawing of a *' single-torn spiral " or of an equilateral triangle, the i *" CH. IX. §3] FIRST PRINCIPLES 123 and readily grasped, I mean both the postulate and the axiom ; but the postulate bids us contrive and find some subject-matter (v\v) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious things ^" Again, says Proclus, '' some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on /ft- equilibrium^ with the remark ' I postulate that equal weights at equal distances are in equilibrium,' though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstration'." " Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subjects" The third view of the distinction between a postulate and an axiom is that of Aristotle above described*. The difficulties in the way of reconciling Euclid's classification of postulates and axioms with any one of the three alternative views are next dwelt upon. If we accept the first view according to which an axiom has reference to something known, and a postulate to something done, then the 4th postulate (that all right angles are equal) is not a postulate ; neither is the 5th which states that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles I less than two right angles. On the second view, the assumption that two straight lines cannot enclose a space, "which even now," says Proclus, "some add as an axiom," and which is peculiar to the subject-matter of geometry, like the fact that all right angles are equal, is not an axiom. According to the third (Aristotelian) view, "everything which is confirmed {irurrovraC) by a sort of demonstration spiral requiring more complex machinery and even the equilateral triangle needing a certain method. ** For the geometrical intelligence will say that by conceiving a straight line fixed at one end but, as regards the other end, moving round the fixed end, and^a point moving along the straight line from the fixed end, I have described the single-turn spiral ; for the end of the straight line describing a circle, and the point moving on the straignt Une simul- taneously, when they arrive and meet at the same point, complete such a spiral. And again, if I draw equal circles, join their common point to the centres of the circles and draw a I straight line from one of the centres to the other, I shall have the equilateral triangle. . These thinsfs then are far from being completed by means of a single act or of a moment's ' thought" (p. 180, 8— ai). , ' Proclus, p. 181, 4— II. * It is necessary to coin a word to render dycovpportwr, which is moreover in the plural. The title of the treatise as we have it is Equilibria of planes or centres tf gravity of planes in Book I and Equilibria of plana in Book ii. » Proclus, p. 181, 16---13. * ibid. p. 183, 6—14. • Pp. 118, 119. 134 INTRODUCnOli fiaiL»t3 will be a postulate, and whaA h incapable of proof wUi be an aacicMii^* This last statement of Produs is h>o8e, as regards the axdom, bacioie it omits Aristotle's requirement that the axiom dioold be a iitf- evident truth, and one that must be admitted by any one who la to learn anything at all, and, as rq^ards the postulate, because Afistade calls a postulate something assumed without proof Ihou^i it is "matter of demonstration" ((kweUtierim Jh% but says nothing ofim ^t^ji-demonstration of the postulates. CHi the whme I think it is from Aristotle that we get the best idea of what Eudid undtnUmd by a postulate and an axiom or common nolioii. Thus ArisMttfs account of an axiom as ai prindpie common to all scienoesi wUdi b self-evident, though incapaUe of proo^ agrees suffidentiy with tfie contents of Euclid's commom naiwms as reduced to five te tiie most recent text (not omitting the fourtii, that ^things whidk oolndd^ are I equal to one another"). As rc^rds the pasimlaies^ it must be borne f in mind that Aristotle says elsewhere* that, ''other things beinff eqQal» that proof is the better whidi proceeds from the fewer postulates or hypotheses or propositions.** If then we say that a geometer must lay down as principles, first certain axicMns or common notions^ and then an irreducible mimm$tm of postulates In tfie Aristolelkui saa«e concerned only with the subject-matter of geometry, we are n6t hr from describing what Euclid in fiurt does. Ais regards the pos^datM we may imagine him sayii^: ^ Besides the common nottons there are a few other things whidi 1 must a^ume without proof, but idiieh differ from the common notioins in that th^ are not sdf«evident The learner may or may not be disposed to agree to them ; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in- the course of the investigation which follows. In the first place certain simple constructions, the drawing and producing of a straight line, and the drawing of a circle, must be assumed to be possible, and with the constructions the existence of such things as straight lines and circles ; and besides this we must lay down some postulate to form the basis of the theory of parallels." It is true that the admission of the 4th postulate that all right angles are equal still presents a difficulty to which we shall have to recur. There is of course no foundation for the idea, which has found its way into many text-books, that *' the object of the postulates is to declare that the only instruments the use of which is permitted in geometry are the ruU and compassK" § 4. THEOREMS AND PROBLEMS. "Again the deductions from the first principles," says Proclus, . "are divided into problems and theorems, the former embracing the ( * Proclus, pp. 18a, 91 — 183, 13. ' Cf. Lardner's Endid : also Todhnnter. • Atta/. poit. I. a5, 86 a 33—35. CH. ix.$4] THEOREMS AND PROBLEMS 125 generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each*." " Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we r^ard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way of theorem and not by way of problem* l^aina Oeo^pfffiarncw aXX* ov wpofiXfffiarucih •* Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish {iropi- aaaOai) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else. ''In reality both assertions are correct Speusippus is right because the problems of geometry are not like those of mechanics, the latter being matters of sense and exhibiting becoming and change of every sort. The school of Menaechmus are right also because the discoveries even of theorems do not arise without an issuing-forth into matter, by which I mean intelligible matter. Thus forms going out into matter and giving it shape may fairly be said to be like processes of becoming. For we say that the motion of our thought and the throwing-out of the forms in it is what produces the figures in the imagination and the conditions subsisting in them. It is in the imagination that constructions, divisions, placings, applications, additions and subtractions (take place), but everything in the mind is fixed and immune from becoming and from every sort of change'." / " Now those who distinguish the theorem from the problem say that every problem implies the possibility, not only of that which is predicated of its subject-matter, but also of its opposite, whereas every theorem implies the possibility of the thing predicated but not of its opposite as well. By the subject-matter I mean the genus which is the subject of inquiry, for example, a triangle or a square or a circle, and by the property predicated the essential attribute, as equality, section, position, and the like, v When then any one * Prodas, p. 77, 7— n. • ML pp. 77, 15—78, 8. » Md. pp. 78, 8—79, a. tt6 INTRODUCTION - t^^«;|4 enunciates thus, To inscrtbt an equilateral triangle in a circle, he states a problem \ for it is also possible to inscribe in it a triangle which is not equilateral Again, if we take the enunciation On a given limited straight line to construct an equilateral triangle^ this is a problem ; for it is possible also to construct one which is not equi- lateral. But, when any one enunciates that In isosceles triangles the angles at the base are equal, we must say that he enunciates a theorem ; for it is not also possible that the angles at the base of isosceles triangles should be unequal. It follows that, if any one were to use the form of a problem and say In a semkircle to describe a right angie^ he would be set down as no geometer For every angle in a semi* circle is rights" " Zenodotus, who belonged to the succession of Oenopides, but was a disciple of Andron^ distinguished the theorem from the problem by the fact that the theorem inquires what is the property predicated of the subject-matter in it, but the problem what is the cause of what effect {rivo^ ivro^ ri ianv}. Hence too Fosidonius defined the one (the problem) as a proposition in which it is inquired whether a thing exists or not (ct ctniv ^ fuj), the other (the theorem') as a proposition in which it is inquired what (a thing) is or of what nature (ri iarw ^ woUv Ti) ; and he said that the theoretic proposition must be put in a declaratory form, eg., Any iHangk has two sules (together) greater than the remaining side and /n any isosceles triangle the angles at the base are equals but that we should state the problematic proposition as if inquiring whether it is possible to construct an equilateral triangle upon such and such a straight line. For there is a difference between inquiring absolutely and indeterminately {a'nXm tc xal dopiarm^) whether there exists a straight line from such and such a point at right angles to such and such a straight line and investigating which is the straight line at right angles V ** That there is a certain difference between the problem and the theorem is clear from what has been said ; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases^ 'that which it was required to do/ and in others, ' that which it was required to prove/ the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction : for we bring in the demonstration in order to show that what was enjoined has been done ; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first i ^ Prodns, pp.79, II— 80,5- f * In the text we have t6 di wftfthiiuL answering torhith withoot snbstantive : wphfikmuk . was obviously inserted in enor. • Froclua, pp. 80. 15—81, 4. CH. IX. §4] THEOREMS AND PROBLEMS 127 book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theorems V* Again, in his note on Eucl. I. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, *' says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things ? But the enunciation of a theorem Ls a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fall short of the truth ; an example is found even in this proposition (I. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed ; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since ' up to our time,' says Carpus, * no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple.' After these distinctions, he proceeds: 'Hence it is that in the Elements too problems precede theorems, and the Elements begin from them ; the first theorem is fourth in order, not because the fifth* is proved from the problems, but because, even if it needs for its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demon- stration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedence'.'" Proclus himself explains the position of Prop. 4 after Props, i — 3 as due to the fact that a theorem about the essential properties of triangles ought not to be introduced before we know that such a thing as a triangle can be constructed, nor a theorem about the equality of sides or straight lines until we have shown, by constructing them, that there can be two straight lines which are equal to one another^ It is plausible enough to argue in this way that Props. 2 and 3 at all events should precede Prop. 4. And Prop, i is used in * Proclus, p. 81, 5 — M. * rh rifiwTow, This should apparently be the fourth because in the next words it is implied that none of the first three propositions are required in prorii^ it. * Proclus, pp. 141, 19 — 143, II. * ibid, pp. 333, ai— 134, 0. 138 INTRODUCTION [CB. n. f 4 Prop. 2, and must therefore precede it But Prop, i showing how to construct an equilateral triangle on a given base is not important, in relation to Prop. 4, as dealing mth the ** production of triangles " in general : for it is of no use to say, as Proclus does, that the construc- tion of the equilateral triangle is "common to the three species (of triangles)\'' as we are not in a position to know this at such an early stage. The existence of triangles in general was doubtless assumed as following from the existence of straight lines and points in one plane and from the possibility of drawing a straight line from one point to another. Proclus does not however seem to reject definitely the view of Carpus, for he goes on* : ** And perhaps problems are in order before theorenis, and especially for those who need to ascend from the arts which are concerned with things of sense to theoretical investigation. But in dignity theorems are prior to problems.... It is then focuish to blame Geminus for saying that the theorem is more perfect than the problem. For Carpus himself gave the priority to problems in respect of order^ and Geminus to theorems in point of more perfect eSgrnty^ so that there was no real inconsistency between the twa Problems were classified according to the number of their possible solutions. Amphinomus said that those which had a unique solution (uorax»?) were called ''ordered'' (the word has dropped out in Proclus, but it must be rvt9tiyA»a^ in contrast to the tfiird kind» iraicr€L)\ those which had a ddfinite number of solutions "inter- mediate " (jkkfra) ; and those with an infinite variety of solutions " un- ordered" (£Taicra)^ Proclus gives as an example of the last tihe problem To divide a given straight line into three farts in continued proportion^. This is the same thing as solving the equations X'\'y-\'Z^a^ xz «>*. Proclus' remarks upon the problem show that it was solved, like all quadratic equations, by the method of '* application of areas." The straight line a was first divided into any two parts, (x-Vz) and j^, subject to the sole limitation that {x-Vz) must not be less than 2y^ which limitation is the SiopurfjLo^, or condition of possibility. Then an area was applied to (;r+ir), or'(a~'y\ ^'falling short by a square figure" (fiCKelirov elSei rerpaywv^) and equal to the square on y. This determines x and z separately m terms of a and y. For, if ir be the side of the square by whiph the area (i.e. rectangle) " falls short," we have {(a --y) -z]z ->•, whence 2z « (a -y) ± •J [{a -yY - 4j/^}. And y may be chosen arbitrarily, provided that it is not greater than a/3. Hence there are an infinite number of solutions. U y^aji, then, as Proclus remarks, the three parts are equal. Other distinctions between different kinds of problems are added by Proclus. The word " problem," he says, is used in several senses. In its widest sense it may mean anything " propounded " (^/Kn-ciyd- fuvov\ whether for the purpose of instruction (jLoBrjatm) or construc- tion (^oii}atfa>9). (In this sense, therefore, it would include a theorem.) ^ Prodns, p. 934, 91. * ibU. p. 143, 19—95. * ihid, p. 990^ 7 — 19. ^ ibid. pp. 990^ 16—991, 6. I CH. ix.§4] THEOREMS AND PROBLEMS 139 But its special sense in mathematics is that of something *' propounded with a view to a theoretic construction'." Again you may apply the term (in this restricted sense) even to something which is impossible, although it is more appropriately used of what is possible and neither asks too much nor contains too little in the shape of data. According as a problem has one or other of these defects respectively, it is called (i) a problem in excess (irXeovd^op) or (2) a deficient problem (jKKvirh irpopkruia). The problem in excess (i) is of two kinds, {a) a problem in which the properties of the figure to be found are either inconsistent {aavfifiara) or non-existent (avvTrap/era), in which case the problem is called impossible, or (b) a problem in which the enunciation is merely redundant : an example of this would be a problem requiring us to construct an equilateral triangle with its vertical angle equal to two-thirds of a right angle ; such a problem is possible and is called "more than a problem" (fiei^op ^ irpofikfffia). The deficient problem (2) is similarly called " less than a problem " (IXaar.aop fj wp6fi\fffia), its characteristic being that something has to be added to the enunciation in order to convert it from indeterminateness (aopurria) to order (rof w) and scientific deter- minateness (Spo^ hriarrffiopuco^) : such would be a problem bidding you " to construct an isosceles triangle," for the varieties of isosceles triangles are unlimited. Such ''problems" are not problems ki the proper sense {Kvplw^ Xeyofiepa wpol3Xf}fiaTa), but only equivocally". § 5. THE FORMAL DIVISIONS OF A PROPOSITION. "Every problem," says Proclus', "and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (wporaai^), setting-out (licOeai^), definition or specification {SiopuTfi6^\ construction or machinery (tearaa-icevi^), proof (dwo&eifi^), conclusion (avfiwipao'fia). Now of these the enunciation states what is given and what is that which is sought, the perfect enunciation consisting of both these parts. The setting-out marks off what is given, by itself, and adapts it before- hand for use in the investigation. The definition or specification states separately and makes clear what the particular thing is which is sought. The construction ox machinery adds what is wanting to the datum for the purpose of finding what is sought. The pro^ draws the required inference by reasoning scientifically from acknowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all , are enunciation, proof, conclusion. For it is equally necessary to know beforehand what is sought, to prove this by means of the intermediate steps, and to state the proved fact as a conclusion ; it is impossible to dispense with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. ^ Proclus, p. an, 7 — 11. ■ ihid, pp. an, 13 — aaa, 14. • ibid. pp. ao3, 1—104, 13 ; 104, a3— «05» 8. O H. E. 130 INTRODUCTION [ch. «• 1 5 Thus there is neither settiftg-ami nor defittUum in the problem of constructing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting-out suffices without any addition for proving the required property from the data. When then do we say that the setting-cut is wanting ? The answer is, when there is nothing given in the enunciation \ for, though the enunciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what b sought, i.e. what must be known or found, as in the case of the problem just mentioned. That problem does not, in (act, state befordumd | with what datum we are to construct the isosceles triangle having j each of the equal angles double of the remaining angle, but (simply) | that we are to find such a triangle.... When, then, the enuncia- tion contains both (what is given and what is sought), in that case we find both definition and setting-out^ but, whenever the datum is wanting, they too are wanting. For not only is the settit^-out concerned with the datum, but so is the definition also^ as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to? But that is what the enundaiioH stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enumiation!^ The constituent parts of an Euclidean proposition will be readily identified by means of the above description. As regards the defi- nition or specification (Biopicfio^) it is to be observed that we have here only one of its uses. Here it means a closer definition or descrip- tion of the object aimed at, by means of the concrete lines or figures set out in the exOeai^ instead of the general terms used in the enun- ciation ; and its purpose is to rivet the attention better, as Proclus indicates in a later passage {rpoirov rwh irpoaex^la^ iarlv alrio^ o htopiaiUs^y. The other technical use of the word to signify the limitations to which the possible solutions of a problem are subject is also described by Proclus, who speaks of Biopia^ioi determining ''whether what is sought is impossible or possible, and how far it is practicable and in how many ways'" ; and the Biopurfio^ in this sense appears in Euclid as well as in Archimedes and Apollonius. Thus we have in Eucl. I. 22 the enunciation *'From three straight lines which are equal to three given straight lines to construct a triangle," followed imme- « diately by the limiting condition (Bioptafi6<;). "Thus two of the straight lines taken together in any manner must be greater than the remaining one." Similarly in VI. 28 the enunciation "To a given straight line to apply a parallelogram equal to a given rectilineal > Proclus, p. 108, II. * Mii. p. 303, 3. CH. IX. is] FORMAL DIVISIONS OF A PROPOSITION 131 figure and falling short by a parallelogrrammic figure similar to a given one " is at once followed by the necessary condition of possi- bility: "Thus the given rectilineal figure must not be greater than that described on half the line and similar to the defect." Tannery supposed that, in giving the other description of the hiopwiU^ as quoted above, Proclus, or rather his g^ide, was using the term incorrectly. The SiopiafAo^ in tlie better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might. Tannery thought, be due to a confusion occasioned by the use of the same words {Set £17) in introducing the parts of a proposition corresponding to the two meanings of the word Siopur/AOf;^ On the other hand it is to be observed that Eutocius distinguishes clearly between the two uses and implies that the differ- ence was well known*. The Biopiafiif: in the sense of condition of possibility follows immediately on the enunciation, is even part of it ; the Biopiafio^ in the other sense of course comes immediately after the Proclus has a useful observation respecting the concltision of a proposition'. "The conclusion they are accustomed to make double in a certain way : I mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial con- clusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a' view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the par- ticular things set out, not qud particulars, but qud typical of the rest For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. . If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of recti- lineal angle ; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general." ' La Ciomitrie grecque^ p. 149 note. Where det 9^ introduces the closer description of the problem we may translate, "it is then required** or '*thus it is required" (to construct etc): when it introduces the condition of possibility we may translate **thus it is necessary etc.*' Heiberg originally wrote dct M in the latter sense in I. 13 on the authority of Produs and Eutocius, and against that of the Mss. Later, on the occasion of xi. 33, he observed that he should have followed the mss. and written dct ^ which he found to be, after all, the right reading in Eutocius (Apollonius, ed. Heiberg, 11. p. 178). det ^ is also the expression usied by Diophantus for introducing conditions of possibility. ' See the passage of Eutocius referred to in last note. ' Proclus, p. 107, 4 — 15. 9—2 132 INTRODUCTION [cb. dl |6 § 6. OTHER TECHNICAL TERMS. I. Things said to be given. Proclus attaches to his description of the formal ^ivisiona of a proposition an explanation of the different senses in which the word given or datum (BeSofiit^ov) is used in geometry. ''Everything that is given is given in one or other of the following ways^ in fosiiiam^ in ratio, in magnitude, or in specks. The point is given in pasiium only, but a line and the rest may be given in all the senses^" The illustrations which Proclus gives of the four senses in which a thing may be given are not altogether happy, and, as regards things which are given in position^ in fn^gnitude^ and in species, it is best, I think, to follow the definitions given by Euclid himself in his book of Data. Euclid does not mention the fourth class, things given in ratio^ nor apparently do any of the great eeometers. ( 1 ) Given in position really needs no definition ; and, when Euclid says (Data, Def 4) that ^Points, lines and angles are said to ht given in position which always occupy the same place,| we are not really the wiser. (2) [Given in magnitude is defined thus {Data^ De£ l): ''Aieas, lines and angles are called given in tnagnitude to which we can find equals."! Proclus' illustration is in this case the following: when, he says, two unequal straight lines are given from the greater of which we have to cut ofT a straight line equal to the lesser, & straight lines are obviously given in magnitude^ *' for greater and less, and finite and infinite are predications peculiar to magnitude." But he does not explain that part of the implication of the term is that a thing is given in magnitude only, and that, for example, its position is not given and is ajnatter of indifference. j(3) Given in species. Euclid's definition {Data, Def. 3) is: '' Rectilineal figures are said to b^ given in species in which the angles are severally given and the ratios of the sides to one another are given/' I And this is the recognised use of the term (cf. Pappus, passim). Proclus uses the term in a much wider sense for which I am not aware of any authority. Thus, he says, when we speak of (bisect- ing) a given rectilineal angle, the angle is given in species by the word rectilineal, which prevents our attempting, by the same method, to bisect a curvilineal angle ! On Eucl. i. 9, to which he here refers, he says that an angle is given in species when e.g. we say that it is right or acute or obtuse or rectilineal or " mixed," but that the actual angle in the proposition is given in species only. As a matter of fact, we should say that the actual angle in the figure of the proposition is given in magnitude and not in species, part of the implication of given in species being that the actual magnitude of the thing given in species is indifferent ; an angle cannot be given in species in this sense at a^. The confusion in Proclus' mind is shown when, after saying that a right angle is given in species, he describes a third of a right angle as given in magnitude. ' Prodiu, p. «05, 13-^15. CH. ix.§6] OTHER TECHNICAL TERMS 133 No better example of what is meant by given in species^ in its proper sense, as limited to rectilineal figures, can be quoted than the g^ven parallelogram in Eucl. VI. 28, to which the required parallelo- gram has to he made similar; the former parallelogram is in fact given in species^ though its actual size, or scale, is indifferent (4) I Given in ratio presumably means something which is given by means of its ratio to some other given thing.( This we gather from Proclus' remark (in his note on I. 9) that an angle may be given in ratio ** as when we say that it is double and treble of such and such an angle or, generally, greater and less." The term, however, appears to have no authority and to serve no purpose. Proclus may have derived it from such expressions as "in a given ratio" which are common enough. 2. Lemma. **The term Umma" says Proclus*, "is often used of any proposition which is assumed for the construction of something else : thus it is a common remark that a proof has been made out of such and such lemmas. But the special meaning of lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itself and therefore worthy of investigation, we call it a lemma\ differing as it does from the postulate and the axiom in being matter of demonstration, whereas they are immediately taken for granted, without demonstration, for the purpose of confirming other things. Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method ; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery. * Proclus, pp. ail, i — an, 4. * It would appear, sajrs Tannery (p. 15111.), that Geminus understood a lemma as being simply XcMi/Sar^eror, something assumed (cf. the passage of Proclus, p. 73, 4, relating to Menaechmus' view of eiements) : hence we cannot consider ourselves authorised in attributing to Geminus the more technical definition of the term here given by Proclus, according to which it b only used of propositions not proved beforehand. This view of a lemma must be considered as relatively modem. It seems to have had its origin in an imperfection of method. In the course of a demonstration it was necessary to assume a proposition which required proof, but the proof of which would, if inserted in the particular place, break the thread ot the demonstration : hence it was necessary either to prove it beforehand as a preliminary proposition or to postpone it to be proved afterwards {in i^ 5ctx^cra«). When, after the time of Geminus, the progress of oneinal discovery in geometry was arrested, geometers occupied themselves with the study and elucidation of the works of the great mathematicians who had preceded them. This involved the investigation of propositions explicitly quoted or tsicitly assumed in the great classical treatises; and naturally it was found that several such remained to be demonstrated, either because the authors had omitted them as being easy enough to be left to the reader himself to prove, or because books in irhich they were proved had been lost in the meantime. Hence arose a class of complementary or auxiliary propositions which were called iemptas. Thus Pappus gives in his Book vn a collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the so-called "Treasury of Analysis " (r6rot dvaXu^erot). When Proclus goes on to distinguish three methods of discovering lemmas, analysis^ dnnsicn, and reductic ad adsurditmf he seems to imply that the principal business of contemporary geometers was the investigation of these auxiliary propositions. Iu~ 134 INTRODUCTION [CB. n. |6 Nevertheless certain methods have been handed down. The finest b the method which by means of attafysis carries the thing aought up to an acknowledged principle, a method which Plato, as tney say. ccMn- municated to Leodamas\ and by which the latter, too, is flAid to have discovered many things in geometry. The second is tiie method of division^ which divides into its parts the genus proposed for con- sideration and gives a starting-point for the demonstration b^ means of the elimination of the other elements in the construction of what is proposed, which method also Plato extolled as being of assistance to all sciences. The third is that by means of the ndMctw ad aisunbim^ which does not show what is sought directly, but refutes its opposite and discovers the truth incidentaUy.** 3. Case. "^ The case^ (wrScif:)" Proclus proceeds^ ''announces difierent ways of construction and alteration <^ positions due to the transposition of points or lines or planes or solidk And, in general, all its varieties are seen in the figure, and this is why it b called our, being a trans- position in the construction." 4. Porism. ** The term porism is used also <^ certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some otiher theorem b revealed at the same time without our propounding it which theorem has on thb very account been called a porism (corollary) as being a sort of inddenttd gain arising from the scientific demonstration*." Cf the note on L 15. * This punge and another from Dioeenes Laertias (ill. 34, p. 7^ ed. Cobet) to the cflect that *' He [Plato] explained (tlnrrh^^r^ to Lcodamas of Thasos the method of inquiry by analysis " have been commonly understood as ascribing to Plato the invention of the method of analvsts ; but Tannery points out forcibly (pp. 1 ta, 113) how difficult it is to explain in what Plato's discovery could have consisted if tmalfsis be taken in the sense attributed to it In Pappus, where we can see no more than a series of successive reductions of a problem until It is finally reduced to a known problem. On the other hand, Proclus' words about • carrying up the thing sought to *' an acknowledged principle " suggest that what he had in mind was the process described at the end of Book vi of the Republic by which the dialec< tidan (unlike tne mathematician) uses hypotheses as stepping-stones np to a principle which is not hypothetical, and then is able to descend step Dy step verifying every one of the hypotheses by which he ascended. This description does not of course rder to mathematical analvsis, but it may have given rise to the ioea that analysis was Plato's discovery, since anafytis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intdlectual method. And it may be that Plato's achievement was to observe the importance, from the pomt of view of logiod rigour, of the confirmatory synthesis following analysis, and to regularise in this way and elevate Into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended. * Here apin the successive bipartitions of genera into species such as we find in the Sophist and Republic have very little to say to geometry, and the very fact that they are hoe mentioned side bv side with analysis suggests that Proclus confused the latter with the philosophical method of Ref. vi. * Tannery rightly remarks (p. 15a) that the subdivision of a theorem or problem Into several cases is foreign to the really classic form ; the ancients preferred, where necessary, to multiply enunciations. As, however, some omissions necessarily occurred, the writers of lemmas naturally added separate casa, which in some instances found their way into die text. A good example is Eudid i. 7, the second case of which, as it appears in our text-books, was interpolated. On the commentary of Proclus on this proposition Th. Taylor rightly remarks that *' Euclid everywhere avokfs a multitude of cases." * Produs, p. SIS, 5— II. * Tannery notes however that, so fin'^lrom distingutshing his corollaries from the con- CH. ix.§6] OTHER TECHNICAL TERMS 135 5. Objection. " The objectuni (Ivaraa-i^) obstructs the whole course of the argu- ment by appearing as an obstacle (or crying ' halt/ ehravTwaa) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything : on the contrary it is necessary to destroy the objection and to show that its author is saying what is false*." That is, in general the objection endeavours to make it appear that the demonstration is not true in every case ; and it is then necessary to prove, in refutation of the objection, either that the supposed case is impossible, or that the demonstration is true even for that case. A good instance is afforded by Eucl. I. 7. The text- books give a second case which is not in the original text of Euclid. Proclus remarks on the proposition as given by Euclid that the objection may conceivably be raised that what Euclid declares to be impossible may after all be possible in the event of one pair of straight lines falling completely within the other pair. Proclus then refutes the objection by proving the impossibility in that case also. His proof then came to ht given in the text-books as part of Euclid's proposition. The objection is one of the technical terms in Aristotle's logic and its nature is explained in the Prior Analytics^ "An objectiofi is a proposition contrary to a proposition.... Objections are of two sorts, general or partial.... For when it is maintained that an attribute belongs to every (member of a class), we object either that it belongs to none (of the class) or that there is some one (member of the class) to which it does not belong." 6. Reduction. This is again an Aristotelian term, explained in the Prior Analytics^ It is well described by Proclus in the following passage : " Reduction {dwayaryi]) is a transition from one problem or theorem to another, the solution or proof of which makes that which is pro- pounded manifest also. For example, after the doubling of the cube had been investigated, they transformed the investigation into another upon which it follows, namely the finding of the two means ; and from that time forward they inquired how between two given straight lines two mean proportionals could be discovered. And they say that the first to effect the reduction of difficult constructions was Hippocrates of Chios, who also squared a lune and discovered many other things in geometry, being second to none in ingenuity as r^ards constructions*." dnsioDS of his propositions, Euclid inserts them before the closing words "(being) what it was required to do*' or "to prove.** In fact the porism-corollary is with Euclid rather a modifira form of the regular conclusion than a separate proposition. * Proclus, p. 211, 18 — 13. ■ ^na/. /nor. ii. 26, 69 a 37. ' idid, II. 35, 69 a 10. * Proclus, pp. 113, 14—313, II. This passage has frequently been taken as crediting Hippocrates with the discovery of the method of geometrical reduction : cL Taylor (Transla- tion of Proclus, II. p. 16), Allman (p. 41 n., 59), Gow (pp. 169, 170). As Tannery remarks (p. 110), if the particular reduction of the duplication problem to that of the two means is 13d INTROftUCtlOK {^ hu ii 7, Reductio ad absurdum. This is variously called by Aristotle "rediuiia ad absurdum^' (^ m nJ m^aTov awaywy ri)\ *" proof /^r impossibik'^ {i} Zia rov aSvudrau Bei^i^i or airoSfif*?)*, "proof leading to the impossible" {^ ca9 to tiBvparoif ayov^a fhraBeih^y. It is part of *' proof (starting) from a hypothesis**' (cf vwo0€iJ€oi^). "All (syllogisms) which reach the conclusion fier imfossibili reason out a conclusion which is false, and they prove the original contention (by the method starting) from a hypothesis, when something impossible results from assuming the contradictory of the original contention, as, for example, when it is proved that the diagonal (of a square) is incommensurable because, if it be assumed commensurable, it will follow that odd (numbers) are equal to even (numbers)*/* Or again, *' proof (leading) to the impossible differs from the direct (S€*<t*«^5) in that it assumes what it desires to destroy [namely the hypothesis of the falsity of the conclusion] and then reduces it to something admittedly false, whereas the direct proof starts from premisses admittedly true'.'* Proclus has the following description of the reductw ad absurdum. '* Proofs by reductio ad ahsurdum \n every case reach a conclusion manifestly impossible, a conclusion the contradictory of which is admitted. In some cases the conclusions are found to conflict with the common notions, or the postulates, or the hypotheses (from which we started) ; in others they contradict propositions previously estab- lished^".*/' Every reducth ad absurdum assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity, and^ by thus destroying the hypothesis, establishes the result originally desired. For it is necessary to under- stand generally that all mathematical arguments either proceed from the first principles or lead back to them, as Porphyry somewhere says. And those which proceed from the hrsX principles are again of two kinds, for they start cither from common notions and the clearness of the self-evident alond or from results previously proved \ while those which lead back to the principles are either by way of assuming the principles or by way of destroying them. Those which assume the principles are called analyses^ and the opposite of these are sjnfAeses — for it is possible to start from the said principles and to proceed in the regular order to the desired conclusion, and this process is syn- ihesis — while the arguments which would destroy the principles are the first noted in hisioryj it « difficult to £tipf>o*ic that it was really the firet ; for Hippocrates x&u&l have found instances of ii in the Pythi^ortjji geomctiy- lirelschneidei, I ihiiiL, tonics nearer the truth when he boldly (p. 99) translates: "Tms reduction c/tie t/artsaiii ram- struct ion is said to have been fint given by Hippocrates.'* The words are wfitrntf Ik fun xAr Aropavfidwm BtaypofiMrif rV lva7«ryV voci^o^^t which must, literally, be translated as in the text above; but, when Proclus speaks vaguely of "difficult oonstractioiis,'' he probably means to say simply that '* this firrt recorded instance of a reduction of a difficult construction is attributed to Hippocrates."' I Aristotle, Anai.frwr, i. 7t ^9 b 5 ; i. 44, 50 a 30. * Uid. I. 91, 39 b 51 ; I. ap, 45 a 55. * Anai, past. I. 94, 85 a 10 etc. ^ Anal, prior, i. 93, 40 b 95. * Anal. prUr. I. 93, 41 a 94. * iHd. 11. 14, 69 b 99. ^ Produs, p. 954, 99—97. r CH. IX.46] OTHER TECHNICAL TERMS 137 called reductiones ad absurdum. For it is the function of this method to upset something admitted as clear*." 8. Analysis and Synthesis. It will be seen from the note on Eucl. XIII. i that the MSS. of the Elements contain definitions of Analysis and Synthesis followed by alternative proofs of xill. i — 5 after that method. The definitions and alternative proofs are interpolated, but they have great historical interest because of the possibility that they represent an ancient method of dealing with these propositions, anterior to Euclid. The propositions give properties of a line cut "in extreme and mean ratio," and they are preliminary to the construction and comparison of the five regular solids. Now Pappus, in the section of his CoUection dealing with the latter subject*, says that he will give the comparisons between the five figures, the pyramid, cube, octahedron, dodecahedron and icosahedron, which have equal surfaces, " not by means of the so-called analytical vdK^VTf, by which some of the ancients worked out the proofs, but by the synthetical method*...." The conjecture of Bretschneider that the matter interpolated in Eucl. xill. is a survival of investiga- tions due to Eudoxus has at first sight much to commend it^ In the first place, we are told by Proclus that Eudoxus " greatly added to the number of the theorems which Plato originated regarding the section^ and employed in them the method of analysis'.** It is obvious that " the section " was some particular section which by the time of Plato had assumed great importance ; and the one section of which this can safely be said is that which was called the " golden section," namely, the division of a straight line in extreme and mean ratio which appears in Eucl. II. 1 1 and is therefore most probably Pytha- gorean. Secondly, as Cantor points out*, Eudoxus was the founder of the theory of proportions in the form in which we find it in Euclid v., VI., and it was no doubt through meeting, in the course of his investigations, with proportions not expressible by whole numbers that he came to realise the necessity for a new theory of proportions which should be applicable to incommensurable as well as commen- surable magnitudes. The "golden section" would furnish such a case. And it is even mentioned by Proclus in this connexion. He is explaining' that it is only in arithmetic that all quantities bear "rational" ratios (/J»;toc >jlrio^) to one another, while in geometry there are '* irrational " ones (apptfro^) as well. " Theorems about sections like those in Euclid's second Book are common to both [arithmetic and geometry] except that in which the straight line is cut in extreme and mean ratio^** » Proclus, p. 355, 8— «6. ■ Pappus, V. p. 410 sqq. • ibid. pp. 410, 17 — 4H, a. ^ Bretschneider, p. 108. See however Heibeig's recent suggestion (Paralipomena tu EukUd in Hermes^ xxxviii., 1003) that the author was Heron. The suggestion is based on a comparison with the remarks on analysis and synthesis Quoted from Heron by an-NairizI (ed. Cortxe, p. 89) at the beginning of his commentary on Eucl. Book il. On the whole, this suggestion commends itself to me more than that of Bretschneider. * Proclus, p. 67, 6. * Cantor, Gtsch, d, Maih, if, p. 941. ' Proclus, p. 60, 7 — 9. • ibid, p. 60, 16 — 19. 138 INTRODUCTION [cB.a.f6 The definitions of Analysis and Symthssis interpolated in EucL XIII. are as follows (I adopt the reading of B and V, the only in- telligible one, for the second^ " Analysis is an assumption of that which is sought as if it were admitted < and the passage > through its consequences to sbmething admitted (to be) true. " Synthesis is an assumption of that which is admitted < and the passage > through its consequences to the finishing or attainment of what is sought" The language is by no means clear and has, at the best, Id be filled out. Pappus has a fuller account* : "* The so-called avakuiiktya^ (' Treasury of Analysis ') is. to put it shortly, a special body of doctrine provided for the use of those who. after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the woric of three men, Euclid the author of the Elements, ApoUonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis. " Analysis then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (tctoi^X ^^'^ ^"^ inquire what it is from which this results,^ and again what is the ante- cedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards {avdiraKiv \vcivy, " But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought ; and this we call synthesis. '' Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematicaL (i) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted : then {a\ if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but {b\ if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted : if then {a) what is admitted is possible and obtainable, that is, what mathematicians call given^ what was originally proposed will also be possible, and the proof will again correspond in * PApptts, vii. pp. 634 — 6. il CH.ix.§6] OTHER TECHNICAL TERMS 139 reverse order to the ianalysis, but if (d) we come upon something admittedly impossible, the problem will also be impossible." The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen ; others by Ofterdinger and Cantor should also be mentioned^ The method is as follows. It is required, let us say, to prove that a certain proposition A is true. We assume as a hypothesis that A is true and, starting from this we find that, if A is true, a certain other proposition B is true ; if B is true, then C ; and so on until we arrive at a proposition K which is admittedly true. The object of the method is to enable us to infer, in the reverse order, that, since K is true, the proposition A originally assumed is true. Now Aristotle had already made it clear that false hypotheses might lead to a conclusion which is true. There is therefore a possibility of error unless a certain precaution is taken. While, for example, B may be a necessary consequence of A, it may happen that A is not a necessary consequence of B. Thus, in order that the reverse inference from the truth of K that A is true may be logically justified, it is necessary that each step in the chain of inferences should be unconditionally convertible. As a matter of fact, a very large number of theorems in elementary geometry are unconditionally convertible, so that in practice the difficulty in securing that the successive steps shall be convertible is not so great as might be supposed. But care is always necessary. For example, as Hankel says', a proposition may not be uncon- ditionally convertible in the form in which it is generally quoted. Thus the proposition " The vertices of all triangles having a common base and constant vertical angle lie on a circle " cannot be converted into the proposition that "All triangles with common base and vertices lying on a circle have a constant vertical angle*'; for this is only true if the further conditions are satisfied (i) that the circle passes through the extremities of the common base and (2) that only that part of the circle is taken as the locus of the vertices which lies on one side of the base. If these conditions are added, the proposition is unconditionally convertible. Or again, as Zeuthen remarks', K may be obtained by a series of inferences in which A or some other proposition in the series is only apparently used ; this would be the case e.g. when the method of modem algebra is being employed and the expressions on each side of the sig^ of equality have been inadvenently multiplied by some composite magnitude which is in reality equal to zero. Although the above extract from Pappus does not make it clear that each step in the chain of argument must be convertible in the case taken, he almost implies this in the second part of the definition of Analysis where, instead of speaking of the consequences B, C... ^ Hankel, Zur Gtschichte der Mathematik in Altertkum und MUUlalter^ 1 874, pp. 137— 1 50 ; Duhamel, Dts mStkodes dans Us sciences de raisonnement^ Part I., 3 ed., Paris, 1885, pp. 39 — 68 ; 2^then, Gesckickie der Mathematik im Altertum und Mitteiaiter^ 18961 pp. 91 — 104; Ofterdinger, Beitrage tmr Gesckickie der grieckiscken Maikimaiik^ Ulm, 1800; Cantor, Gesckickie der Matkematik, ij, pp. a 10 — 1. ' Hankel, p- 139. ' Zeuthen, p. 103. I40 INTRODUCTION [CB. OL f 6 successively following from A» he suddenly changes the expreitkm and says that we inquire wkai ii is (B)fivm which A fallows (A bdng thus the consequence of B, instead of the reverse), and then what (viz. C) is the antecedent cause of B; and in practice tlie Greeks secured what was wanted by always insisting on the analysis being confirmed by subsequent synthesis, that is, they laboriously worked backwards the whole way from K to Ap reversing the order of the analysis, which process would undoubtedly bring to light any flaw which had crept into the argument through tibe accidental neglect of the necessary precautions. Reductio ad absurdum a variety of analysis. In- the process of analysis starting from the hypothesis that a proposition A is true and passing through B, C... as successive con- sequences we may arrive at a proposition K which, instead of being admittedly true, is either admittedly false or the contradictory of the original hypothesis A or of some one or more of the propositions B^ C... intermediate between A and K. Now correct inference from a true proposition cannot lead to a false ^proposition ; and in this case there- fore we may at once conclude, wiuout any inquiry whether the various steps in the ailment are convertible or not, that the hypo- thesis A is false, for, if it were true, all the consequences correctly inferred from it would be true and no incompatibility could arise. This method of proving that a given hypothesis is false furnishes an indirect method of proving that a given hypodiesis A is true^ since we have only to take the contradictory of A and to prove that it is false. This is the method of reductio ad absurdum^ which is therefore a variety of analysis. The contradictory of A, or not- A, will generally include more than one case and, in order to prove its falsity, each of the cases must be separately disposed of: e.g., if it is desired to prove that a certain part of a figure is equal to some other part, we take separately the hypotheses (i) that it is greater^ (2) that it is less^ and prove that each of these hypotheses leads to a conclusion either admittedly false or contradictory to the hypothesis itself or to some one of its consequences. Analysis as applied to problems. It is in relation to problems that the ancient analysis has the greatest significance, because it was the one general method which 3ie Greeks used for solving all "the more abstruse problems" (rii daa^arepa' rAv TrpofiXsffLdrc^py. I We have, let us suppose, to construct a figure satisfying a certain set of conditions. If we are to proceed at all methodically and not by mere guesswork, it is first necessary to "analyse" those conditions. To enable this to be done we must get them clearly in our minds, which is only possible by assuming all the conditions to be actually fulfilled, in other words, by supposing the problem solved. Then we have to transform those conditions, by all the means which practice in such cases has taught us to employ, into other conditions which are necessarily fulfilled if the original conditions are, and to continue this ^ Prodiu, p. 94*, 16, 17. CH. ix.§6] OTHER TECHNICAL TERMS 141 transformation until we at length arrive at conditions which we are in a position to satisfy ^ In other words, we must arrive at some relation which enables us to construct a particular part of the figure which, it is true, has been hypothetically assumed and even drawn, but which nevertheless really requires to h^ found in order that the problem may be solved. From that moment the particular part of the figure becomes one of the data, and a fresh relation has to be found which enables a fresh part of the figure to be determined by means of the original data and the new one together. When this is done, the second new part of the figure also belongs to the data ; and we proceed in this way until all the parts of the required figure are found*. The first part of the analysis down to the point of discovery of a relation which enables us to say that a certain new part of the figure not belonging to the original data is given, Hankel calls the transformation ; the second part, in which it is proved that all the remaining parts of the figure are "given," he calls the resolution. Then follows the synthesis^ which also consists of two parts, (i) the construction^ in the order in which it has to be actually carried out, and in general following the course of the second part of the analysis, the resolution ; (2) the demonstration that the figure obtained does satisfy all the given conditions, which follows the steps of the first part of the analysis, the transformation, but in the reverse order. The second part of the analysis, the resolution, would be much facilitated and shortened by the existence of a systematic collection of Data such as Euclid's ^ book bearing that title, Jbonsisting of propositions proving that, if in a figure certain parts or relations arc given, other parts or relations are also given(/ As regards the first part of the analysis, the trans- formation, the usual rule applies that every step in the chain must be unconditionally convertible; and any failure to observe this condition will be brought to light by the subsequent synthesis. The second part, the resolution, can be directly turned into the construction since that only is given which can be constructed by the means provided in the Elements. It would be difllicult to find a better illustration of the above than the example chosen by Hankel from Pappus.^ Given a circle ABC and two joints D, E external to it, to draw straight lines DB, l£.Efrom D,E to a point B on the circle such that, if DB, IL^ produced meet the circle again in C, A, AC shcdl be parallel to DE. Analysis. Suppose the problem solved and the tangent at A drawn, meeting ED produced in F. (Part I. Transformation.) Then, since AC is parallel to D£, the angle at C is equal to the angle CDE. But, since FA is a tangent, the angle at C is equal to the angle FAE. Therefore the angle FAE is equal to the angle CDE, whence A, By D, F are concyclic. * 2^then, p. 93. • Hankel, p. 141. ' » Pappus, vii. pp. 830— «. 143 INTRODUCTION I [CH. UL |6 I Therefore the rectangle AE^ EB is equal to tiie rectan^e FR^ ED. (Part II. Resolution.) But the rectangle AE^ EB is given, because it is equal to the square on the tangent from E. Therefore the rectangle FE^ ED is given; and, since ED is given, FE is nven (in length). [Data, 57J But FE is given in position also,- so that F is also given. [Daia^ 27.] Now FA is the tangent from a given point F to a circle ABC given in position ; therefore FA is given in position and magnitude. [Daia^ 9a] And F is given ; therefore A is given. But E is also given ; therefore the straight line AE is given in position. [Dttia^ 26.] And the circle ABC is given in position ; therefore the point B is also given. [Data^ 35.] But the points D, E are idso given ; therefore the straight lines DB^ BE are also given in position. Synthesis. (Part I. Construction.) Suppose the circle ABC and the points A E given. Take a rectangle contained by ED and by a certain strai^t line EF equal to the square on the tangent to the circle from E. From F draw FA touching the circle in A ; join ABE and then DB^ producing DB to meet the circle at C. Join A C. 1 say then that AC is parallel to DE. (Part II. Demonstration.) Since, by hypothesis, the rectangle FEy ED is equal to the square on the tangent from Ey which again is equal to the rectangle AEy EB^ the rectangle AEy EB is equal to the rectangle FEy ED. Therefore AyByDyF are concyclic, whence the angle FAE is equal to the angle BDE. But the angle FAE is equal to the angle ACB in the alternate segment ; therefore the angle A CB is equal to the angle BDE. Therefore AC\s parallel to DE. In cases where a Siopurfio^ is necessary, i.e. where a solution is only possible under certain conditions, the analysis will enable those ] conditions to be ascertained. Sometimes the Siopiafio^ is stated and proved at the end of the analysis, e.g. in Archimedes, On the Sphere and Cylindery II. 7 ; sometimes it is stated in that place and the proof postponed till after the end of the synthesis, e.g. in the solution of the problem subsidiary to On tfte Sphere and Cylindery II. 4, preserved in Eutocius' commentary on that proposition. The analysis should also enable us to determine the number of solutions of which the problem is susceptible. ' CH. ix.§7] THE DEFINITIONS 143 § 7. THE DEFINITIONS. General. " Real " and " Nominal " Definitions. It is necessary, says Aristotle, whenever any one treats of any whole subject, to divide the genus into its primary constituents, those which are indivisible in species respectively: e.g. number must be divided into triad and dyad ; then an attempt must be made in this way to obtain definitions, e.g. of a straight line, of a circle, and of a right angle'. The word for definition is 2/)09. The original meaning of this word seems to have been "boundary," "landmark." Then we have it in Plato and Aristotle in the sense of standard or determining principle ("id quo alicuius rei natura constituitur vel definitur," Index AristotelicusY ; and closely connected with this is the sense of definition. Aristotle uses both 8/309 and opurfio^ for definition, the former occurring more frequently in the Topics, the latter in the MetapAj^sics. Let us now first be clear as to what a definition does not do. There is nothing in connexion with definitions which Aristotle takes more pains to emphasise than that a definition asserts nothing as to the existence or non-existetice ol the thing defined. It is an answer to the question what a thing is (r/ ^cm), and does not say tliat it is (oTi itrrC). The existence of the various things defined has to be proved, except in the case of a few primary things in each science, the existence of which is indemonstrable and must be assumed among the first principles of each science ; c.g[. points and lines in geometry must be assumed to exist, but the existence of everything else must be proved. This is stated clearly in the long passage quoted above under First Principles'. It is reasserted in such passages as the following. "The (answer to the question) what is a man and the fact that a man exists are different things*.*' " It is clear that, even according to the view of definitions now current, those who define things do not prove that they exist'." "We say that it is by demonstration that we must show that everything exists, except essence (c* ft^ ovala tlri). But the existence of a thing is never essence; for the existent is not a genus. Therefore there must be demonstration that a thing exists. Thus, what is meant by triangle the geometer assumes, but that it exists he has to prove V "Anterior knowledge of two sorts is necessary : for it is necessary to presuppose, with regard to some things, that they exist \ in other cases it is necessary to understand what the thing described is, and in other cases It is necessary to do both. Thus, with the fact that one of two contradictories must be true, we must know that it exists (is true); » Anal, past 11. 13. b 15. * Cf. De aninia, 1. 1, 404 a 9, where ** breathing " is spoken of as the 6pot of ** life," and the many passages in the Politics where the wora is used to denole that which gives its special character to the several forms of government (virtue being the tpot of aristocracy, wealth of oligarchy, liberty of democracy, 1194 a 10) ; Plato, Republic^ viii. 551 c. ' Anal. post. I. 10, 76 a 31 sqq. ^ ibid. Ii. 7, 91 b 10. * ibid. 91 b 19. * ibid. 91 b 11 sqq. . 144 INTRODUCTION [CB. n. f 7 of the triangle we must know that it means such and such a tiling ; of the unit we must know botii what it means and that it exists^" What is here so much insisted on is the very fact which Mill pointed out in his discussion of earlier views of Definitions, where he says that the so-called real definitions or definitions of tkin^^ do not constitute a different kind of definition from nominal definitions, or definitions of names ; the former is simply the latter pirns something else, namely a covert assertion that the thii^ defined exists. ''This covert assertion is not a definition but a postulate. The definition is a mere identical proposition which gives information only aboiit the use of language, and from which no conclusion affecting matters of fact can possibly be drawn. The accompanying postulate, on the other hand, affirms a fact which may lead to consequences of evenr degree of importance. It affirms the actual or possible existence of Things possessing the combination of attributes set forth in the definition : and this, if true, may be foundation sufficient on which to build a whole fabric of scientific truth'." This statement really adds nothing to Aristotie*s doctrine': it has even the slight disadvantage, due to the use of the word "postulate" to describe "the covert assertion* in all cases, of not definitely pointing out that there are cases where existence has to be proved as distinct from those where it must be assmmd. It is true that the existence of a definiend may have to be taken for granted provisionally until the time comes for proving it; but, so far as r^ards any case where existence must be proved sooner or later, the provisional assumption would be for Aristotie, not a postulate, but a hypothesis. In modem times, too. Mill's account of the true distinction between real and nominal definitions had been fully anticipated by Saccheri*, the editor of Euclides ab omni naroo vvtdicatus (1733), famous in the history of non-Euclidean geometry. In his Logica Demofistrativa (to which he also refers in his Euclid) Saccheri lays down the clear distinction between what he calls de- finitiones quid nominis or nominaleSy and definitiones quid rei or reales^ namely that the former are only intended to explain the meaning 1 AfuU.posL I. I, 71 a II sqq. ' Mill's System 0/ Logic, Bk. i. ch. viii. < It is true that it was in opposition to ** the ideas of most of the ArisMelian hgidans** (rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his observation: ** We have already made, and shall often have to repeat, the remark, that the philosophers who overthrew Realism by no means got rid of the consequences of Realism, but retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had oeen handed down from Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This, so long as a definition was considered to be a proposition • nnfokling the nature of the thing,* did well enough. But Hobbes followed and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name ; yet he continued to affirm as broadly as any of his predecessors that the d^oZ, frincipia, or original premisses of mathematics, and even of all science, are definitions ; producing the singular paradox that systems of scientific truth, nay, all truths whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words.** ' Aristotle was guilty of no such paradox ; on the contrary, he exposed it as plainly as did Mill. ^ This has been fuDy brought out in two papers by G. Vailati, La i§cria ArisMdica ddim dkAmitione (Rwista di FiUsofia e uienu amm\ 1003)1 and Di tm* opera dtmen tu a /a del P. Gereiamo Saccheri (** Logica DemonstraUva,*' 1(^7) (in Rivista FUiofica^ I903)- f r CH. ix,§7] THE DEFINITIONS 145 that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of constructing it The definitio quid nominis becomes a definitio quid rei " by means of a postulate^ or when we come to the question whether the thing exists and it is answered affirmatively ^^ Definitiones quid nominis are in them- selves quite arbitrary, and neither require nor are capable of proof; they are merely provisional and are only intended to be turned as quickly as possible into definitiones quid rei, either (i) by means of a postulate in which it is asserted or conceded that what is defined exists or can be constructed, e.g. in the case of straight lines and circles, to which Euclid's first three postulates refer, or (2) by means of a demonstration reducing the construction of the figure defined to the successive carrying-out of a certain number of those elementary constructions, the possibility of which is postulated. Thus definitiones quid rei are in general obtained as the result of a series of demonstrations. Saccheri gives as an instance the construction of a square in Euclid I. 46. Suppose that it is objected that Euclid had no right to define a square, as he does at the beginning of the Book, when it was not certain that such a figure exists in nature; the objection, he says, could only have force if, before proving and making the construction, Euclid had assumed the aforesaid figure as given. That Euclid is not guilty of this error is clear from the fact that he never presupposes the existence of the square as defined until after i. 46. Confusion between the nominal and the real definition as thus de- scribed, i.e. the use of the former in demonstration before it has been turned into the latter by the necessary proof that the thing defined exists, is according to Saccheri one of the most fruitful sources of illusory demonstration, and the danger is greater in proportion to the "complexity" of the definition, i.e. the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible, i.e. the simultaneous presence of which in a given figure can be proved, by means of other postulates etc. forming part of the I basis of the science, to be impossible. The same thought is expressed by Leibniz also. " If," he says, " we give any definition, and it is not clear from it that the idea, which we ascribe to the thing, is possible, we cannot rely upon the demon- strations which we have derived from that definition, because, if that idea by chance involves a contradiction, it is possible that even con- I tradictories may be true of it at one and the same time, and thus our ■ demonstrations will be useless. Whence it is clear that definitions I are not arbitrary. And this is a secret which is hardly sufficiently known'." Leibniz' favourite illustration was the " regular polyhedron with ten faces," the impossibility of which is not obvious at first sight. ^ ** Definitio quid nominis nata est evadere definitio auid rei per postulaium vel dum j Yenitur ad quaestionem an est et respondetur affirmative. I * O^cuUs eifragmenis inJdits de Leibniz, Paris, Alcan, 1903, p- 431* Quoted by Vailati. ! H. B. 10 146 INTRODUCTION [CB. dl 1 7 It need hardly be added that, speaking generally, Euclid's defini- tions, and his use of them, agree with the doctrine of Aristotle that the definitions themselves say nothing as to the existence of die things defined, but that the existence of each of them most be pro>^ or (in the case of the ** prindple^'*) assumed. In geometry, says Aristotle, the existence of points and lines only must be as* sumed, the existence of the rest being proved. Accordingly Euclid's first three postulates declare the possmility of constructing straight lines and circles (the only "lines except straight lines UMd in the Elements). Other thin^ are defined and afterwards constructed and prov«l to exist : e.g. in cook L, Def. 20, it is explained what is meant by an equilateral triangle ; then (L i ) it is proposed to construct it, and, when constructed, it is proved to agree with the definitioa When a square is defined (l. De£ 22), the question whether such a thing really exists is left open until, in 1. 46, it is proposed to construct it and, when constructed, it is proved to satisfy the definition^ Similarly with the right angle (L Def. 10, and L ii) and parallels (L De£ 23, and I. 27 — 29). The greatest care is taken to exclude mere presumption and imagination. The transition from the sub- jective definition of names to the objective definition of things is made, in geometry, by means of coHStmctions (the first principles of lyhich are postulated), as in other sciences it is made by means ctf . experience*. } Aristotle's requirements in a definition. We now come to the positive characteristics by which, according to Aristotle, scientific definitions must be marked. Firsts the different attributes in a definition, when taken separately, cover more than the notion defined, but the combination of them i does not Aristotle illustrates this by the " triad," into which enter the several notions of number, odd and prime, and the last " in both its two senses (a) of not being measured by any (other) number (ck M^ fierpelaOai dpi0/i£) and (A) of not being obtainable by adding numbers together" (o>9 fi)f avyxeurOai i( dpiSfA&v), a unit not being a number. Of these attributes some are present in all other odd numbers as well, while the last [primeness in the second sense] \ belongs also to the dyad, but in nothing but the triad are they aU present •." The fact can be equally well illustrated from geometry. Thus, e.g. into the definition of a square (Eucl. L, Def. 22) there enter j the several notions of figure, four-sided, equilateral, and right-angled, | each of which covers more than the notion into which aU enter ks attributes! Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things defined*. This is ^ Trenddenbarg, EUmmta Le^ices Aristotdeai^ | 5a * Trendelenbufg, Erldutenmgm tu den EUfnentm der arisUtelisckgn Lagik^ 3 ed. p. io7. On constniction as proof of existence in ancient geometry cf. H. G. ZcaXlktskt Die getmeiriselU Construction als ** ExUtenubem^s ** $n der amtUen GeonutrU (in Matkanatisckg Annalen, 47. Band). * Anai.past, u. 13, 96 a 33— b i. ^ Trendelenbnig, ErUhtieruneent p. 108. * Tsfia vi. 4, 141 a 96 iqq. CH. IX.J7] THE DEFINITIONS 147 clear, since the object of a definition is to give us knowledge of the thing defined, and it is by means of things prior and better known that we acquire fresh knowledge, as in the course of demonstrations. But the terms " prior " and " better known " are, as usual susceptible of two meanings; they may mean (i) absolutely or logically prior and better known, or (2) better known relatively to us. In the absolute sense, or from the standpoint of reason, a point is better known than a line, a line than a plane, and a plane than a solid, as also a unit is better known than number (for the unit is prior to, and the first principle of, any number). Similarly, in the absolute sense, a letter is prior to a syllable. But the case is sometimes different relatively to us ; for example, a solid is more easily realised by the senses than a plane, a plane than a line, and a line than a point. Hence, while it is more scientific to begin with the absolutely prior, it may, perhaps, be permissible, in case the learner is not capable of following the scientific order, to explain things by means of what is more intelligible to him, '^ Among the definitions framed on this principle are £ose of the point, the line and the plane; all these explain what is prior by means of what is posterior, for the point is described as the extremity of a line, the line of a plane, the plane of a solid." But, if it is asserted that such definitions by means of things which are more intelligible relatively only to a particular individual are really definitions, it will follow that there may be many definitions of the same thing, one for each individual for whom a thing is being defined, and even different definitions for one and the same individual at different times, since at first sensible objects are more intelligible, while to a better trained mind they become less so. It follows therefore that a thing should be defined by means of the absolutely prior and not the relatively prior, in order that there may be one sole and immutable definition. This is further enforced by reference to the requirement that a good definition must state the genus and the differentiae, for these are among the things which are, in the absolute sense, better known than, and prior to, the species (j&v cEttXcS^ yvtoptiumipcuv koX irporipcav rov tXSav^ iarlp). For to destroy the genus and the differentia is to destroy the species, so that the former are ^ior to the species ; they are also better known, for, when the species is known, the genus and the differentia must necessarily be known also, e.g. he who knows ** man " must also know " animal " and '' land-animal," but it does not follow, when the genus and differentia are known, that the species is known too, and hence the species is less known than they are^ It may be frankly admitted that the scientific definition will require superior mental powers for its apprehension ; and the extent of its use must be a matter of discretion. So far Aristotle ; and we have here the best possible explanation why Euclid supplemented his definition of a point by the statement in I. Def 3 that the extremities of a line are points and his definition of a surface by I. Def. 6 to the effect that the extremities of a surface are lines. The supplementary expla- * Topics VI. 4, 141 b «5— 34. 148 INTRODUCTION [cH. iz. 1 7 nations do in fact enable us to arrive at a better understanding of the formal definitions of a point and a line respectively, as is well ex* plained by Simson in his note on Def. I. Simson says» namelv. that we must consider a solid, that is, a magnitude wUch has length, breadth and thickness, in order to understand aright the definitions of a point, a line and a surfSsice. Consider, for instance, the boundary common to two solids which are contiguous or the boundary whidi divides one solid into two contiguous parts; this boundary is a surface. We can prove that it has no thickness by taking away either solid, when it remains the boundary of the other; for, if it had thickness, the thickness must either be a part of one solid or of the other, in which case to take away one or otiier solid would take away the thickness and therefore the boundary itself: which is impossible. Thereft)re the boundary or the surface has no thickness. In exactiy the same way, r^arding a line as the boundary of two contiguous surfaces^ we prove that the line has no breadth ; and, lastiy, re^urding a point as the common boundary or extremity of two lines, we prove that a point has no length, breadth or thickness. Aristotie on unscientific definitions. AristoUe distinguishes three kinds of definition which are un- scientific because founded on what b n^/ prior (^^ Ic wfHnipm(% The first is a definition of a 'thin|^ by means of its opposite, eg. of "* good ** by means of " bad " ; this is wrong because opposites are naUirally i evolved together, and the knowledge of opposites is not uiicommonly I regarded as one and the same, so that one of the two opposites '■ cannot be better known than tiie other. It is true that, m some cases of opposites, it would appear that no other sort of definition is possible: e.g. it would seem impossible to define double apart from the half and, generally, this would be the case with things which in their very nature (jcaO' aura) are relative terms {irpi^ n X^ctoa), since one cannot be known without the other, so that in the notion of either the other must be comprised as welP. The second kind of definition which is based on what is not prior is that in which there is a complete circle through the unconscious use in the definition itself of the notion to be defined though not of the name*. Trendelenburg illustrates this by two current definitions, (i) that of magnitude as that which can be increased or diminished, which is bad because the positive and negative comparatives "more" and "less" presuppose . the notion of the positive " great," (2) the famous Euclidean definition ^ of a straight line as that which "lies evenly with the points on itself" (^f laov roU t4^ iavrvf^ ayfieioi^ K€iT(u\ where " lies evenly " can only ^ be understood with the aid of the very notion of a straight line which is to be defined^ The tkird kind of vicious definition from that which is not prior is the definition of one of two coordinate species by means of its coordinate {dvriSinpfffiivov), e.g. a definition of " odd " as that which exceeds the even by a unit (the second alternative in Eud. vii. Def. 7) ; for "odd " and "even " are coordinates, being differentiae of ^ Topia VI. 4, 143 A 13 — 31. * ihid, 143 a 34— b 6. '* Trendelenbuig, BrULwUntngtH^ P* I'S* _^ ) 1 I i CH. IX. § 7] THE DEFINITIONS 149 number^ This third kind is similar to the first. Thus, says Tren- ■f delenburg, it would be wrong to define a square as *'a rectangle \ with equal sides." Aristotle's third requirement. ' A third general observation of Aristotle which is specially relevant ' to geometrical definitions is that "to know what a thing is (rt ifrriv) is the same as knowing why it is (S^ rl iffrip)*.'* " What is an eclipse ? A deprivation of light from the moon through the interposition of the earth. Why does an eclipse take place? Or why is the moon eclipsed ? Because the light fails through the earth obstructing it What is harmony ? A ratio of numbers in high or low pitch. Why does the high-pitched harmonise with the low-pitched? Because the high and the low have a numerical ratio to one another*." *' We seek die cause (to hUtn) when we are already in possession of the fact {to Sri). Sometimes they both become evident at the same time, but at all events the cause cannot possibly be known [as a cause] before the fact is known^'* '' It is impossible to know what a thing is if we do not know that it is*/* Trendelenburg paraphrases : " The definition of the notion does not fulfil its purpose until it is made genetic. It is the producing cause which first reveals the essence of the thing. ••. The nominal definitions of geometry have only a provisional significance and are superseded as soon as they are made genetic by means of construction." Kg. the genetic definition of a parallelogram is evolved from Eucl. I. 31 (giving the construction for parallels) and I. 33 about the lines joining corresponding ends of two straight lines parallel and equal in length. Where existence is proved by construction, the cause and the fact appear together^ Again, *'it is not enough that the defining statement should set forth the fact, as most definitions do; it should also contain and present the cause ; whereas in practice what is stated in the definition is usually no more than a conclusion (avfiiripaa^). For example, {I what is quadrature ? The construction of an equilateral right-angled •I figure equal to an oblong. But such a definition expresses merely the f conclusion. Whereas, if you say that quadrature is the discovery of a i mean proportional, then you state the reason ^" This is better under- i stood if wc compare the statement elsewhere that "the cause is the 'middle term, and this is what is sought in all cases*," and the illustra- ' tion of this by the case of the proposition that the angle in a semi- circle is a right angle. Here the middle term which it is sought to establish by means of the figure is that the angle in the semi-circle is equal to the half of two right angles. We have then the syllogism : Whatever is half of two right angles is a right angle ; the angle in a i semi-circle is the half of two right angles ; therefore {conclusion) the -angle in a semi-circle is a right angle*. As with the demonstration, so * Topia VI. 4, 143 b 7—10. — • Anal, past, 11. 3, 90 a 31. * Anal. post. 11. 3, 90 a 15—31. — * ibid, ii. 8, 93 a 17. * ihid, 9^ a 3a * Trendelenbarg, Erldutemngm, p. 116. ' Dt amma 11. 3, 413 a 13 — 30. * Anal* post. \\. 3, 90 a 6, * Hid. II. II, 94 a 33. 1 ISO INTRODUCTION [cB.ix.|7 it should be with the definition. A definition idiich is to show the genesis of the thing defined should contain the middle term or cause ; otherwise it is a mere statement of a conclusion. Consider, for instance, the definition of "quadrature" as ''making a square equal in area to a rectangle with unequal sides." This gnres no hint as to whether a solution of the problem is possible or how it is solved : bnt, if you add that to find the mean proportional between two given straight lines gives another straight une such that the square on it is | equal to the rectangle contained by the first two straight linesi yoa ■ supply the necessary middle term or cause^. ! Technical terms not defined by Euclid. | It will be observed that what is here defined, ''quadrature* or " squaring " (rerpaywviafAosi), is not a geometrical figure, oran attribute of such a figure or a part of a figure, but a tedmical term used to , describe a certain problem. Euclid does not define such things ; but the fact that Aristotle alludes to this particular definition aswdl as to definitions of deflection {tc€tc\M0ai) and of verging' {vwlkuf) seems to show that earlier text-books included among ddinitions explanations of a number of technical terms, and that Euclid deliberat^ omitted these explanations from his Elements as surplusage. LaCer the tendency was again in the opposite direction, as we see from the mudi expanded Definitions of Heron, which, for example, actually include a definition of a deflected line (xeKkaafUpff ypa^Lia^nr* Eudid uses the passive of kX&p occasionally*, but evidently considered it unneoesraiy to explain such terms, which had come to bear a recognised meaning. The mention too by Aristotle of a definition of verging (yci/tiy) suggests that the problems indicated fa^ this term were not excluded from elementary text-books before Euclid. The type of problem (vevcisi) was that of placing a straight line across two lines, e.g. two straight lines, or a straight line and a circle, so that it shall verge to a given point (i.e. pass through it if produced) and at the same time the intercept on it made by the two given lines shall be of given length. Other posages in Aristotle may be quoted to the like effect : e.g. Anai, pott. i. t, 71 b 9 '* We consider that we know a particniar thing in the absolate sense, as distinct nrom the sophistical and incidental sense, when we consider that we know the cause on aooonnt of which the thing is, in the sense of knowing that it is the canse of that thing and that it cannot be otherwise,*' ibid. i. 13, ^pa 1 '* For here to know ihtfact is the fnnctum of! those who are concerned with sensible thm^ to know the cauu is the ranction of the matbe- , matician ; it is he who possesses the proofi of the causes, and often he does not know the fiict." In view of such passages it is difficult to see how Proclus came to write (p. sos, 11) that Aristotle was the originator ('A^c^rorAow Korif^pmn) of the idea of Amphmomus ana others that geometry does not investigate the cause and the wfy {rh Ml W)- To this Geminus replied that the invertigation of the cause does, on the contrary, appear in geometry. " For how can it be maintained that it is not the business of the geometer to inauire for wluit reaaon, on the one hand, an infinite number of equilateral polygons are inscribea in a circle, but, on the other hand, it is not possible to inscribe in a sphere an infinite numbo' of polyhedral figures, ejquilateral, equiangular, and made up of similar plane figures? Whose business is it to ask this question and find the answer to it if it is not that of the geometer? Now when geometers reason per imfossibiU thejr are content to discover the property, but when they , argue by direct proof, it such pool be only partial (M Mpovf)i thb does not suffice for showing the canse ; if however it is general and applies to all like cases, the why (vt^ M W) is at once and concurrently made evident." * Heron, ed. Hultsch, Def. 14, p. 11. * e.g. in III. 90 and in IkOa 89. IcH. IX.J7] THE DEFINITIONS 151 In general, the use of conies is required for the theoretical solution of I these problems, or a mechanical contrivance for their practical j| solution \ Zeuthen, following Oppermann, gives reasons for supposing, ■ not only that mechanical constructions were practically used by the f older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conies in every case where that was possible*. Heiberg* suggests that the allusion of Aristotle to yevo-ew perhaps confirms this sup- position, as Aristotle nowhere shows the slightest acquaintance with conies. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by " plane " methods (i.e. by means of the straight line and circle). We know, e.g., from Pappus that Apollonius wrote two Books on plane vevaei^*. But one thing is certain, namely that Euclid deliberately excluded this class of Iproblem, doubtless as not being essential in a book of Elements. Definitions not afterwards used. Lastly, Euclid has definitions of some terms which he never after- wards uses, e.g. oblong (irepo/AffKe^), rhombus, rhomboid, trapezium. The "oblong" occurs in Aristotle; and it is certain that all these definitions are survivals from earlier books of Elements. ^ Cf. the chapter on vw&vtit in ne IVarks of Archimedes^ pp. c — cxxii. * Zeuthen, Die Lehre twn den Kegelschniiten im Altertum^ ch. ii, p. 161. ' Heiberg, Mathemaiiseka tu ArisioieUs^ p. 16. * Pi^pns VII. pp. 670—3. 1 ,>■■/ BOOK I. DEFINITIONS. •^ I. A point is that which has no part. <- 2. A line is breadthless length. 3. The extremities of a line are points. — 4. A straight line is a line which lies evenly with the points on itself. — 5* A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal anfi^les is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. J^ii^iacute angle is an angle less than a right angle. — 13. A boundary is that which is an extremity of any- thing. ■^ 14. A figure is* mat which js contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another ; 154 BOOK I [h VIEW. i6— POST. 4 16. And the point is (Ailed the centre of the circle. 1 7. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 1 8. A semicircle is the figure contained by the diameter and the circumference cut off by it . And the centre of the semicircle is the same as that of the circle. 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained j^ four, and multi- lateral those contained by more than four straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. 21. Further, of trilateral figures, a right-angled tri- angle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute- angled triangle that which has its three angles acute. 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. POSTULATES. Let the following be postulated : 1. To draw a straight line from any point to any point 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. -L«. . > ^ I- POST. 5— c N. s] DEFINITIONS ETC 155 5* . That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. COMMON NOTIONS. 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. [7] 4- Things which coincide with one another are equal to one another. [8] 5. The whole is greater than the part Definitio^i I. iilfUUv loTiF, oS fUpot aiOiv, '^'^ A point is that wAick has no part. An exactly parallel use of lUfioi {i<rrC) in the sin^lar is found in Aristotle, Metaph. 1035 b 32 fiipoi fuv cZv Icrrl fcou rov cZSou^ literally *'There is a /jTf/ even of the form "; Bonitz translates as if the plural were used, *'Theile giebt es," and the meaning is simply "even the form is divisiMe (into parts)." Accordingly it would be quite justifiable to translate in this case "A point is that which is indivisible intoparts.^ Martianus Qipella (5th c. a.d.) alone or almost alone translated differently, **Punctum est cuius pars nihil t&\^^ *'a point is that a part of which is nothing,^ Notwithstanding that Max Simon {Eudid und die sechs planimitrischen Bilcher^ 190 1 ) has adopted this translation (on grounds which I shall presently mention), I cannot think that it p^ives any sense. If a part of a point is nothings Euclid Might as well have said that a point is itself ''nothing,'' which of course he does not do. Pre-Euclidean definitions. It would appear that this was not the definition given in earlier text- books; for Aristotle {Topics vi. 4, 141 b 20), in speaking of *7A^ definitions'' of point, line, atod surface, says that they all define the prior by means of the posterior, a point as an extremity of a line, a line of a surface, and a surface of a solid. The first definition of a point of which we hear is that given by the Pythagoreans (cf. Proclus^ o. 95, ai), who defined it as a "monad having position" or ''with position added" (iiova^ wfHxrXaPovau O&riv). It is firequently wed by Aristotle, either in this exact form (cf. De anima L-4^ 409 a 6) or its equivalent: e.g. in Metaph, 7016 b 24 he says that that which is indivisible •very way in respect of ma^ itude and qu& magnitude but has not position is ^wumady while Uiat which is similarly indivisible and has position is 9. paint. Plato ippMR ID IVPO obfected to this definition. Aristotle says (Meti^h. 156 BOOK I [l 993 a 3o) that he objected "to tfab genua [that of pomti]a$ being a geometrical fiction {yu^jutrpudw Uy/ia)^ and caUed a point the beginning m a line (4«^ ypa/ifi'^), while again he fre(}uently npoke of 'indiTiaUe lines."* To wbidi Aristotle replies that even ''indivisil^ lines" most have eitreoiities, so that the same a^ment which proves the existence of Ufus can be used to prove ataX points exist It would appear therefore that, when Aristotle objects to the definition of a point as die extremity of a Une (v^pot ypo^i^) as un- scientific {Topics VI. 4, 141 b 21X he is aiming at Plato. Heiberg conjectures {MathemaHsches mu Aristotdes^ p. 8) diat it was due to Plato's influoice that the word for "point" generally used by Aristode (<myfu|) was r^laoed by tnuMwv (the r^iular term used by Eudid, Archimedes and later wnters), tfie latter term {^nota^ a conventional mark) probably hmg conddeied more suitable than frrvfi»alj (fLpunchtre) whidi mij^ appear to ^lim greater naUiy for a point Anstode's conception of a point as diat which is indivisible and has position is fiirther illustrated by such observations as diat a point is not a body (De caeh 11. 13, 296 a 17) and has no wri^ {ibiJL in. i, S99 a 30); again, we can make no distinction between a point and the/Auy (rmf) where it is {Physics iv. i, 209 a 1 1). He finds the usual difficulty in aooountiii^ for the transition fix>m the indivisible, or infinitely small^ to the finite or divisible magnitude. A point being indivisiUe^ no accumulation of points, however fitf it mav be carried, can give us anything divisiUe, whereas of course a line is a divisible ma^tude. Hence he holds that points cannot make up anythiiig continuous like a line, point cannot be continuous with point (00 yap «Eot&v ixi/uyov arifUiov arifuicv ^ oriyfuf any§KlJ9t -Di gtm» d corr. I. a, 317 a lo), and a line is not made up of points (ov myMtrw cic frttfium^ Pkysia nr. 8, ai^ b 19). A point, he says, is like the mm in time : nam is indivisiUe and is not 9k part of time, it is only the b^;inning or end, or a division, of time, and simikurly a point may be an extremity, b^inning or division of a line, but is not part of it or of magnitude (cf. De cado in. i, 300 a 14, Pf^sia iv. zi, 220 a I — 21, VI. X, 231 b 6 sqq.). It is only by motion that a point can generate a line {De anima i. 4, 409 a 4) and thus be the origin of magnitude. Other ancient definitions. According to an-NauIzi (ed Curtze, p. 3) one "Herundes" (not so fiu: identified) defined a point as "the indivisible beginning of all magnitudes," • and Posidonius as ''an extremity which has no dimension, or an extremity of aline." Criticisms by commentators. Euclid's definition itself is of course practically the same as that which Aristotle's fi'equent aUusions show to have been then current, except that it omits to say that the point must have position. Is it then sufficient, seeing that there are other things which are without parts or indivisible, e.g. the now in time, and the unit in number? Proclus answers (p. 93, 18) that the point is the only thing in the subject-matter of geometry that is indivisible. Relatively therefore to the particular science the definition is sufficient Secondly^ the definition has been over and over again criticised because it is purely n^;ative. Proclus* answer to this is (p. 94, 10) that nq;ative descriptions are appropriate to first principles, and he quotes Pannenide s as having described his first and last cause by means of n^;ations merely. Arif totie too admits that it may sometimes be necessary for one framing a definition to use negations, e.g. in defining privative terms such as "blind"; and he seems to accept as proper ■I L DEF. i] NOTE ON DEFINITION i 157 the negative element in the definition of a point, since he says {De anima iii. 6, 430 b 20) that ''the point and every division [e.g. in a length or in a period of time], and that which is indivisible in this sense, is exhibited as privation Simplicius (quoted by an-NairIzi) says that ''a point is the beginning of magnitudes and that from which they grow ; it is also the only thing wluch, having position, is not divisible.'' He, like Aristotle, adds that it is by its motion that a point can generate a magnitude : the particular magnitude can only be "of one dimension," viz. a line, since the point does not "spread itself" (dimittat). Simplicius further observes that Euclid defined a point n^atively because it was arrived at by detaching surface from body, line from surface, and finally point from line. "Since then body has three dimensions I It follows that a point [arrived at after successively eliminating all three dimensions] has fwne of the dimensions^ and has no part." This of course reappears m modern treatises (cf. Rausenbeiger, Eiementar-geometrie des Punktes^ der Geraden und der Ebtne^ 1S87, p. 7). An-NairizI adds an interesting observation. " If any one seeks to know the essence of a point, a thing more simple than a line, let him, in the sensible world, think of the centre of the universe and the poUs,^^ But there is nothing new under the sun : the same idea is mentioned, in an Aristotelian treatise, in controverting those who imagine that the poles have some influence in the motion of the sphere, "when the poles have no magnitude but are extremities and points" (De motu animalium 3, 699 a 21). Modem views. In the new geometry represented by the excellent treatises which start from new systems of postulates or axioms, the result of the profound study of the fundamental principles of geometry during recent years (I need only mention the names of Pasch, Veronese, Enriques and Hilbert), points come before lines^ but the vain effort to define them a priori is not made ; instead of this, the nearest material things in nature are mentioned as illustrations, with the remark that it is from them that we can get the abstract idea. Cf. the full statement as regards the notion of a point in Weber and Wellstein, Encyclopddit der elementaren Mathematik^ 11., 1905, p. 9. "This notion is evolved from the notion of the real or supposed material point by the process of limits, Le. by an act of the mind which sets a term to a series of presen- tations in itself unlimited. Suppose a grain of sand or a mote in a sunbeam, which continually becomes smaller and smaller. In this way vanishes more and more the possibility of determining still smaller atoms in the grain of sand, and there is evolved, so we say, with growing certainty, the presentation of the point as a definite position in space which is one and is incapable of further division. But this view is untenable ; we have, it is true, some idea how the grain of sand gets smaller and smaller, but only so long as it remains just visible; after that we are completely in the dark, and we cannot see or imagine the further diminutiop. That this procedure comes to an end is unthinkable ; that nevertheless diere exists a term beyond which it cannot go, we must believe or postulate without ever reaching it . . . It is a pure act of wili^ not of the understanding." Max Simon observes similarly (Euclid^ p. 25) " The notion * point ' belongs to the limit-notions (Grenzbegriffe), the necessary conclusions of continued, and in themselves unlimited, series of presentations." He adds, "The point is the limit of localisation; if this is more and more energetically continued, it leads to the limit-notion 'point,' 158^ BOOK I ' [i. DiTP. I, s better 'position,' which at the tame time uivohFa a chai^ Content of space vanishes, relative pasiUm remains. 'Pdnt' theny aocoiding to our interpretation of Euclid, is the eztremest limit of diat iriiidi we can £il think of (not observe) as a i;^^/^ presentation, and if we go further than that^ not only does extension cease but even relative /Auy^ and in this lenie tfie 'put' is nothing.*^ I confess I think that even the meanii^ which Simon intend! to convey is better expressed by ''it has 110 part" than uf "the part is nothing,* since to take a ''part" of a thing in Eudid's tense ii the result of a simple division, corresponding to an arithmetical fraction, would not be to change the notion from that of the thing divided to an entirely diflTerent < f Definition 2. TpofLfi^ Sk firJKOi airXarcc A line is breadihUss length. This definition may safely be attributed to the Platonic School, if not to Plato himself. Aristotle {Topies vl 6, 143 b 11) speaks of it as open to objection because it " divides the goius by n^^ation," loigth being neonsaiily either breadthless or possessed ofbfeadUi; it woukl seem however that tfie objection was only taken in order to score a point against the Phtonists, ainoe he says {ibid. 143 b 29) that the aigument is "of service mtfy against dioae who assert that the genus [sc. length! is one numerically, diat is^ thoae who assume ideas^^* e.g. &e idea of lengta (oAri ^i^Kot) which they r^^ard as a genus : for if the genus, being one and self-existent, could be divided into two species, one of which asserts what the other denies, it would be sdf- contradictory (Waitz). -Proclus (pp. 96, 21 — 97, 3) observes that, whereas the definition of a point is merely ne^tive, the line introduces the first " dimension," and so its definition is to this extent positive^ while it has also a n^;ative element which denies to it the other "dimensions" (Suurrao-ctf). The negation of both breadth and depth is involved in the single expression "breadthless" (onrXarcg), since everything that is without breadth is also destitute of depth, though the converse is of course not true. Alternative definitions. The alternative definition alluded to by Produs, ficycfct c^* tr tMurrftror " magnitude in one dimension " or, better perhaps, " magnitude extended one way " (since fiuurrao-ic as used with reference to line, sur&ce and solid scarcdy corresponds to our use of "dimension" when we speak of "one,'' "two," or "three dimensions"), is attributed by an-NairIzi to "Heromides/* who must presumably be the same as "Herundes," to whom he attributes a certain definition of a point It appears however in substance in Aristotie, though AristoUe does not use the adjective Suurrardp, nor does he apparenUy use Suunroo-if except of body as having three " dimensions " or " having dimension (or extension^ oi/ways (vdb^)," the "dimensions" being in his view (i) up and down, (2) before and behind, and (3) right and left, and "up" being the principle or beginning of lengthy *< ri^ht " of breadth^ and " before " of d^th (De caelo ii. 2, 284 b 24). A line is, according to Aristotle, a magnitude ^divisible in one way only" (fun^xv ^Mupcro^X ^ contrast to a ma^tude divisible in two ways (&X9 ficoi^ror), or a surface, and a magnitude divisible "in all or in three wa^s" {^riirqi ttal rpi)^ iiaip€r6y\ or a body {Meiaph. 1016 b 25 — 27); or It is a magnitude **cmtintkms one way (or in one directionX" as compared with magnitudes continuous two ways or thne ways. ^ I. DEF. a] NOTES ON DEFINITIONS i, a 159 which curiously enough he describes as ** breadth " and "depth" respectively (/Acyctfos 8^ rh fuv i^t* tv ovKcxcf fi'^KOSf rh 8* ^l Suo irX<£ro9, ro 8* ^i rpia piBo^, Metaph, loao a 11), though he immediately adds that "length " means a line, " breadth " a surface, and " depth " a body. Proclus gives another alternative definition as ^^flux of a point '^ (fiwrn fnujuiiov), i.e. the path of a point when moved. This idea is also alluded to in Anstotle (De anima i. 4, 409 a 4 above quoted) : " they say that a line by its motion produces a surface, and a point by its motion a line." "This definition," says Proclus (p. 97, 8 — 13), "is a perfect one as showing the essence of the line : he who called it the flux of a point seems to define it from its genetic cause, and it is not every line that he sets before us, but only the immaterial line ; for it is this that is produced by the point, which, though itself indivisible, is the cause of the existence of things divisible." Proclus (p. 100, 5 — 19) adds the usefiil remark, which, he says, was current in the school of ApoUonius, that we have the notion of a line when we ask for the length of a road or a wall measured merely as length ; for in that case we mean something irrespective of breadth, viz. distance in one " dimension." Further we can obtain sensible perception of a line if we look at the division between the light and the dark when a shadow is thrown on the earth or the moon ; for clearly the division is without breadth, but has length. Species of ** lines." After defining the "line" Euclid only mentions one species of line, the straight line, although of course another species appears in the definition of a circle later. He doubtless omitted all classification of lines as unnecessary for his purpose, whereas, for example. Heron follows up his definition of a line by a division of lines into (i) those which are " straight " and (2) those which are not, and a further division of the latter into {a) "circular circumferences," (b) "spiral-shaped" (IXucofiScis) lines and {c) "curved" (fcofiirvAat) lines generally, and then explains the four terms. Aristotle tells us {Metaph. 986 a 25) that the Pythagoreans distinguished straight (cv^ and curved (fcofiirvAov), and this distinction appears in Plato (cf. Republic x. 602 q) and in Aristotle (cf. " to a line belong tiie attributes straight or curved," AnaL post. i. 4i 73 b 19; "as in mathematics it is useful to know what is meant by the terms straight and curved," De anima i. i, 402 b 19). But firom the class of "curved" lines Plato and Aristotle separate off the vcpi^cpiTs or " circular " as a distinct species often similarly contrasted with straight Aristotle seems to recognise broken lines forming an angle as one line : thus "a line, if it be bent (kcko/a- ficn7), but yet continuous, is called one" {Afetaph, 1 01 6 a 2); "the straight line is more one than the bent line" (ibid. 1016 a 12^. Cf. Heron, Def. 14, "A broken line («cc«cXa<rficn7 ypofifiif) so-called is a luie which, when produced, does not meet itse^.** When Proclus says that both Plato and Aristotle divided lines into those which are "straight," "circular" (ircpi^c/n;^) or "a mixture of the two," adding, as regards Plato, that he included in the last of these classes " those which are caUed helicoidal among plane (curves) and (curves) formed about solids, and such species of curved lines as arise from sections of solids" (p. 104, i — 5), he appears to be not quite exact The reference as regards Plato seems to be to Parmenides 145 b: "At that rate it would seem that the one must have shape, either straight or round (arrpoyyuXov) or some combination of the two"; but this scarcely amounts to a formal classification of lines. As regards i6o BOOK I [h i«r. t Aristotle, Produs seems to have in mind the pasage (Ik mcfr i. 9» t68 b i^) where it is stated that ''all maiiom in qpace, which we call translatioa (4^p£^ is (in) a straight line, a circle, or a combmation of the two; for the first two are the only simple (motions)** For completeness it is desirable to add the substance of Produs* account of the classification of lines, for frtiich he quotes Geminus as his authority. Geminus' first classification of lines. This bqi;ins (p. 1 1 1, i — o) with a division of lines into tmt^osik (otMctoc) and incomposite (dirvi^cTot). The only illustration given of the iomforiU class is the *' broken line which forms an angle** (^ «cicXa<r/&^ rai yn way voiovo-a) ; the subdivision of the huomposiii class then follows (in the test as it stands the word " composite " is clearly an error for *' incomposite % The subdivisions of the incompcMite class are repeated in a later passage (pp- 176, 27 — 177, 23) with some additional details. The following diagram reproduces the effect of both versions as fiu: as possible (all the illustrations mentioned Iqr Produs being shown in brackets). Unet oonipotitc iooonipotite (broken line fbnmng an ana^) I forming a figure or determinate (drde, ellipse, dssoid) not forming a fignie or indeCenninate extending witboot limit (itraia;ht Une, parabola, nypeibola. I) The additional details in the second version, which cannot easily be shown in the diagram, are as follows : (i) Of the lines which extend without limit, some do not/arm a figure at aU (viz. the straight line, the parabola and the hyperbola); but some first *'come together and form a figure ** (Le. have a loop), ''and, for the rest, extend witibout limit" (p. 177, 8). As the only other curve, besides the parabola and the hyperbola, which has been mentioned as proceeding to infinity is the conchoid (of Nicomedes), we can hardly avoid the condusion of Tannery^ that the curve which has a loop and then proceeds to infinity is a variety of the conchoid itself. As is ^ Noiapomr tkistoire dis lignes it surf oca ccmrbes dans ftuUiquHi in BnUetim dts sciem^ mathhn, dastronom. t i6r. viii. (1884), PP* '08— 9. ■j L DEF. a] NOTE ON DEFINITION a x6i well known, the ordinary conchoid (which was used both for doubling the cube and for trisecting the angle) is obtained in this way. Suppose any number of rays passing through a fixed point (the pole) and intersecting a fixed straight line ; and suppose that points are taken on the rays, beyond the fixed straight line, such that the portions of the rays intercept^ between the fixed straight line and the point are equal to a constant distance (iidtmifia), the locus of the points is a conchoid which has the fixed straight line for asymptote. If the "distance " a is measured fi-om the intersection of the ray with the given straight line, not in the direction awa^ from the pole, but towards the pole, we obtain three other curves accordmg as a is less than, equal to, or greater than ^, the distance of the pole from the fixed straight line, which is an asymptote in each case. The case in which a>d gives a curve which forms a loop and then proceeds to infinity in the way Proclus describes. Now we know both from Eutocius (Comm. an Archimedes^ ed. Heiberg, iii. p. 114) and Proclus (p. 272, 3 — 7) that Nicomedes wrote on conchoidr (in the plural) and Pappus (iv. p. 244, 18) says that besides the "first" (used as above stated) there were "the second, the third and the fourth which are useful for other theorems.'' (2) Proclus next observes (p. 177, 9) that, of the lines which extend without limit, some are ^^ asymfioiic** (davfwrTWToi), namely "those which never meet, however they are produced," and some are ^^ symptotic^** namely "those which will meet sometime"; and, of the "asymptotic" class, some are in one plane, and others not Lastly, of the "asymptotic" lines in one plane, some preserve always the same distance firom one another, while others continually "lessen the distance, like the hyperbola with reference to the straight line, and the conchoid with reference to the straight line." ^ Geminus' second classification. I This (from Proclus, pp. iii, 9 — 20 and 112, 16—18) can be shown in a diagram thus : Incomposite lines I dffit^BiTOi ypa/iiuU f I ' I ' simple, irXfj mixed, fuicHj 1 . I making a figure indeterminate (e.g. circle) (straight line) I ' 1 lines in planes lines on solids I td h roit CT€ptott n line meeting itself extending without limit (e.g. cissoid) lines formed by sections lines round solids aX Karii rdt roftdt al w€pl rd rr€p€d (e.g. conic sections, spiric curves) (e.g. heiix about a sphere or about a cone) (cylindrical helix) (all others) Notes on classes of "lines" and on particular curves. We will now add the most interesting notes found in Proclus with reference to the above classifications or the particular curves mentioned. H. E. II L x6a BOOK I [i. DV. t 1. Homoeomeric lines. By this tenn {ifunofuptU) are meant lines which are alike in all partii so that in any one such curve any part can be made to coincide with any odier part Proclus observes that these lines are only three in number, two being ''simple'' and in a plane (the stra^ht line anid the circle^ and the dura ''mixed," (subsisting) "alxMit a sohd,** namdy the cylindrical hdiz. The latter curve was also called the nxkiias or tufckUom^ and its k^moemmnc property was proved by ApoUonius in his woric wiyn tov K^xkiom (Prochis, p. 105, 5). The fact that mere are only three kamoeomiru lines was proved by Geminus, "who proved, as a preliminary proposition, that, if from a point {M Tov o^ficiov, but on p. 251, 4 iu^ Mt oiyfMiov) two stnuf^t lines be orawn to a homoeomeric line making equal angles with it, the straight Hnei are equal" (pp. 112, 1— 113, 3. cfc P- ^S't 2—19)- 2. Mixed lines. It might be supposed, says Proclus (p. 105, xxX that the cylindrical hdiz, being homoeomeric^ like the straight liiMS and the cirde, must like them be simple. He replies that it is not smiple, but mixei^ because it is generated by two unlike motions. Two like motions, said Geminus, eg. two motions at the same speed in the directions of two adjoining sides of a sc^uare, produce a simple hne, namely a straight line (the diagonal); and again, if a straight line moves with its extremities upon the two sides of a right an^e respectively, i this same motion gives a simpk curve (a circle) for tte locus of the midme J point of the straight line, and a mUxii curve (an dlipae) for the locus of any I other point on it (p. 106, 3 — 15). i Geminus also explained that the term ^ mixed," as ai^lied to curves, and as applied to surfaces, respectively, is used in different senses. As apf^ed to | curves, "mixing" neither means simple "putting together ** ((nMco-it) nor " blending " (Kpcuric). Thus the helix (or spiral) is a " mixed " Une, but (i) it j is not " mixed '* in the sense of "putting together," as it would be if, say, part ] of it were straight and part circular, and (2) it is not mixed in the sense of " blending," beoiuse, if it is cut in any way, it does not present the appearance of any simple lines (of which it might be supposed to be compounded, as it \ were). The " mixing " in the case of lines is rather that in which the con- stituents are destroyed so far as their own character is concerned, and are replaced, as it were, by a chemical combination {yrrw iv avrj avpt^apfiiva rk oKpa KoX inrvK€xiffjJva). On the Other hand " mixed " surfaces are mixed in the sense of a sort of " blending " (icara rti^a Kpcuriv). For take a cone gene- ' rated by a straight line passing through a fixed point and passing always j through the circumference of a circle : if you cut this by a plane puallel to ^ that of the circle, you obtain a circular section, and if you cut it by a plane through the vertex, you obtain a triangle, the " mixed " surface of the cone | being thus cut into simple lines (pp. 117, 22 — 118, 23). 3. Spiric curves. These curves, classed with conies as being sections of solids, were dis- covered by Perseus, according to an epigram of Eratosthenes quoted by Proclus (p. 112, i), which says that Perseus found "three lines upon (or, perhaps, m addition to) five sections" (rpcic ypa/ifia« M wivT€ to/muv). Proclus throws some light upon these in the following passages : "Of the spiric sections, one is interfaced, resembling the horse-fetter j {hntov wi&ti) ; another is widened out in the middle and contracts on each 1! I.DEF. a] NOTE ON DEFINITION d 163 side (of the middle), a third is elongated and is narrower in the middle, broadening out on each side of it" (p. 112, 4 — 8). "This is the case with the spiric sutface \ for it is conceived as generated by the revolution of a circle remaining at right angles [to a plane] and turning about a point which is not its centre [in other words, generated by the revo- lution of a circle about a straight line in its plane not passing through the centre]. Hence the spire takes three forms, for the centre [of rotation] is either on the circumference, or within it, or without it. And if the centre of rotation is on the circumference, we have the continuous spire (owfxi;?), if within, the interlaced (ifjLV9ir\€yfuvyj\ and if without, the open (Sicxi/f). And the spiric sections are three according to these three differences'' (p. 119, 8-17). " When the Aippopede, which is one of the spiric curves, forms an angle with itself, this angle also is contained by mixed lines" (p. 127, i — 3). "Perseus showed for spirics what was their property ((rvfiirrcDfui) *' (P- 356, "). Thus the spiric surface was what we call a tore, or (when open) an anchor- ring. Heron (Def 98) says it was called alternatively spire (cnrcipa) or ring (fcpocos); he calls the variety in which "the circle cuts itself," not "interlaced," but "crossing-itself" (ffvaXXarrovo-a). Tannery^ has discussed these passages, as also did Schiaparelli*. It is clear that Proclus' remark that the difference in the three curves which he mentions corresponds to the difference between the three surfaces is a slip, due perhaps to too hurried transcribing from Geminus : all three arise from plane sections of the open anchor-ring. If r is the radius of the revolving circle, a the distance of its centre from the axis of rotation, d the distance of the plane section (supposed to be parallel to the axis) from the axis, the three curves described in the first extract correspond to the following cases : (i) d^a-r. In this case the curve is the hippopede^ of which the lemniscate of Bernoulli is a particular case, namely that in which a = 2r. The name hippopede was doubtless adopted for this one of Perseus' curves on the ground of its resemblance to the hippopede of Eudoxus, which seems to have been the curve of intersection of a sphere with a cylinder touching it internally. i2) fl + r > //><!. Here the curve is an ovaL 3) a>d>a-r. The curve is now narrowest in the middle. Tannery explains the "three lines upon (in addition to) five sections" thus. He points out that with the open tore there are two other sections corresponding to (4) d= a : transition from (2) to (3). (5) fl-r>^>o, in which case the section consists of two symmetrical ovals. He then shows that the sections of the closed or continuous tore^ corre- sponding to a = r, give curves corresponding to (2), (3) and (4) only. Instead of (i) and (5) we have only a section consisting of two equal circles touching one another. On the other hand, the third spire (the interlaced variety) gives three new forms, which make a group of three in addition to the first group oifive sections. > Pour rhistoire da Hgtus et surfaces courbts dams rasttiquitl in BulUtin des sciences mathim, et astronom. vili. (1884), pp. 35 — 37. ' Die kofmoceniriscken Spkarm des Eudoxus^ des Kallippus und des ArisioteUs (Ahhrnnd- lungm tstr Geuk, der Maik. I. Heft, 1877, pp. 149 — 151. II — 2 L-— 164 BOOK I [l. DEP. The difficulty which I see in this interpretation is the fiict that, just after "three lines on five sections " are mentioned, Produs describes three curves which were evidently the roost important ; but these three belong to three of the five sections of the open tore^ and are not separate from them. 4. The cissoid. This curve is assumed to be the same as that by means of which, accordirig to Eutodus {Comm. on Atxkimiies^ in. p. 79 sqq.^ Diodes in his book w^ irvfMiv {On iuming-gkuses) solved the problem of doubling the cube. It is the locus of points which he found b^ the following construction. Let AC^ BD be diameters at right angles in a arcle with centre A Let E^ J^be points on the quacbants BC^ BA respectivdy such that the arcs BE^ BE axe equal. Draw EG, Elf perpendicular to CA. Join AE^ and let F be its intersecticm with Eir. The dssoid is the locus of all the points E corresponding to different posi- tions of E on the quadrant BC and of E at an equal distance from B along the arc BA. ^ is the pointy the curve correspond- ing to the position C for the point JE*, and B the point on the curve correqKmding to the position of E in which it ooinddes with A It is easy to see that the curve extends in the direction AB beyond B, and that CJ^ drawn perpendicular to CA is an asjrmptote. It may be regarded also as having a branch AD symmetrical with ABf and, beyond D^ approaching EC produced as asjrmptote. If OA, OD are coordinate axes, the equation of the curve is obviously where a is the radius of the drcle. There is a cusp at A, and it agrees with this that Proclus should say (p. 126, 24) that "dssoidal lines converging to one point like the leaves of ivy — for this is the origin of their name — form an angle." He makes the slight correction (p. 128, 5) that it is not two parts of a curve, but am curve, which in this case makes an angle. But what is surprising is that Produs seems to have no idea of the curve passing outside the drcle and having an asymptote, for he several times speaks of it as a closed curve (forming a figure and including an area): cf. p. 152, 7, "the plane (area) cut off by the cissoidal line has one bounding (line), but it has not in it a centre such that all (straight lines drawn to the curve) fix>m it are equal." It would appear as if Proclus regarded the cissoid as formed by ^efour symmetrical dssoidal arcs shown in the figure. Even more pecub'ar is Proclus' view of the 5. *• Single-turn Spiral." This is really the spiral of Archimedes traced by a point starting fix>m the fixed extremity of a straight line and moving uniformly along it, while 1 I. DEFF. 2—4] NOTES ON DEFINITIONS 2—4 165 simultaneously the straight line itself moves uniformly in a plane about its fixed extremity. In Archimedes the spiral has of course any number of turns, the straight line making the same number of complete revolutions. Yet Proclus, while giving the same account of the generation of the spiral (p. 180, 8 — 12), regards the single-turn spiral as actually stopping short at the point reached after one complete revolution of the straight line : " it is necessary to knaw that extending without limit is not a property of all lines; for it neither belongs to the circle nor to the cissoid, nor in general to lines which form figures ; nor even to those which do not form figures. For even the single- turn spiral does not extend without MmiX— far it is constructed between two points — nor does any 0/ the other lines so generated do so'* (p. 187, 19 — 25). It is curious that Pappus (viii. p. mo sqq.) uses the same term fiovoa-Tpo^ 2Ai^ to denote one turn, not of the spiral, but of the cylindrical helix. Definition 3. Tpofjififj^ ik wipara irqiiMia, The extremities of a line are points. It being unscientific, as Aristotle said, to define a point as the '* extremity of a line " («pas ypafifi^^)^ thereby explaining the prior by the posterior, Euclid defined a point differently; then, as it was nec^^Bary to connect a point with a line, he introduced this escplanation after the definitions of both had been given. This compromise is no doubt his own idea; the same thing occurs with reference to a surface and a line as its extremity in Def. 6, and with reference to a solid and a surface as its extremity in xi. Def. 2. We miss a statement of the facts, equally requiring to be known, that a ** division " ^Stcupco-ts) of a line, no less than its " begmning ** or *' end," is a point (this is brought out by Aristotle: cf. Jfetaph, 1060 b 15), and that the intersection of two lines is also a point If these additional explanations had been given, Proclus would have been spared the difiSculty which he finds in the fact that some of the lines used in Euclid (namely infinite straight lines on the one hand, and circles on the other) have no "extremities." So also the ellipse, which Proclus calls by the old name Bvptos (" shield "). In the case of the circle and ellipse we can, he observes (p. 103, 7), take a portion I bounded by points, and the definition applies to that portion. His rather I far-fetched distinction between two aspects of a circle or ellipse as a line and * as a closed figure (thus, while you are describing a circle, you have two extremi- < ties at any moment, but they disappear when it is finished) is an unnecessarily elaborate attempt to establish the literal universality of the "definition," which is really no more than an explanation that, if a line has extremities, those extremities are points. Definition 4. A straight line is a line which lies evenly with the points on itself The only definition of a straight line authenticated as pre-Euclidean is that of Plato, who defined it as " that of which the middle covers the ends " (relatively, that is, to an eye placed at either end and looking along the straight line). It appears in the Farmenides 137 s : "straight is whatever has its middle in fi'ont of (Le. so placed as to obstruct the view of) both its ends " x66 BOOK I [i.Dir.4 (cMv y€ ov ivri fuo-ovififi^o&roir J^TOCV.Mvpotf^crf). Alistode ^poCtt it in equivalent terms (72^/W vi. ix, 148 b 2j\ oS ri jUnv iw u rfi 9 $ u row wipmn; and, as he does not mention the name of its author, but states it in combina- tion with the definition of a line as the extremity of a surbce, we vm assume that he used it as being weU known. Produs also quotes the definition as ipiato's in almost identictd termSi' i% ra gUm roUiKpoii9iw a r pa o$u(p. X091 ai). This definition is ingenious, but implicitly appeals to the sense of sight and involves the postulate that the line of sight is straiffht (Cf. the A ri stotel i an Problems 31, 20, 959 a 39, where the question is why we can better observe straightness in a row, say, of letters with one eye than with twa) As nqguds the straightness of ''visual iEys,'*a^cii^ d Euclid's own Q^tks, De£ x, a, assumed as hypotheses^ in which he first speaks of the ''stnuj^t lines" drawn from the eye, avoiding the word jfcis, and then sa^ that the figure contained by the visual rays (o^ces) is a cone with its vertex m the eye« As Aristotle mentions no definition of a straight line resembling EucUd's, but gives only Plato's definition and the other explaining it as the ''extremity of a surface,'' the latter being evidently the current defimtion in contemporary textbooks, we may safely infer that Euclid's definition was a new departure oif his own. Proclus on Euclid's definition. Coming now to the interpretation of Euclid's definition, cMm yni^ iariVf ipif i( wov roi« i^* javr^ mffuioft kuto^ we find any number of sUghtly different versions, but none that can be described as quite satisfiictory ; some authorities, e.g. Savile, have confessed that they could make nothing of it It is natural to appeal to Proclus first ; and we find that he does in net give an interpretation which at first sight seems plausible. He says (p. X09, 8 sq.) that Euchd "shows by means of this that the straight line alone [of all lines] occupies a distance (icarcxciv Su^on^/ia) equal to that between the points on it For, as fiar as one of the points is distant from another, so great is the length (f&^yc^os) of the straight line of which they are the extremities ; and this b the meaning of lying i( urov to (or with) the points on it " \i£ ttrw being thus, J apparently, interpreted as "at" (or "over") "an equal distance"]. "But if \ you take two points on the circumference (of a circle) or any other line, the f distance cut off between 'them along the line is greater than the interval separating them. And this is the case with every line except the straight line. Hence the ordinary remark, based on a common notion, that those who journey in a straight line only travel the necessary distance, while those who do not go straight travel more than the necessary distance." (Cf. Aristotle, j De catlo i. 4, ayi a 13, "we always .call 'the distance of anything the straight 1 line " drawn to it) Thus Proclus would interpret somewhat in this way : "a J straight line is that which represents extension equal with (the distances separating) the points 'on it" This explanation seems to be an attempt to i graft on to Euclid's definition the assumption (it is a Aofi/Savd/icvor, not a definition) of Archimedes {On the sphere and cylinder i. ad init) that "of all < the lines which have the same extremities the straight line is least" For this , purpose li Zcrov has apparently to be tiCken as meaning "at an equal distance," and again "lying at an e<^ual distance" as equivalent to "extending over (or * representing) an equal distance." This is difficult enough in itself, but is seen to be an impossible interpretation when applied to the similar definition of a plane by Euclid (Def. 7) as a surfece "which lies evenly with the straight lines on itself." In that connexion Proclus tries to make the same words If Zirou t J ' I. DEF. 4] NOTE ON DEFINITION 4 167 Mirai mean "extends over an equal area with." He says namely (p. 117, 2) that ''if two straight lines are set out" on the plane, the plane surface '' occupies a space equal to that between the straight lines." But two straight lines do not determine by themselves any space at all ; it would be necessary to have a closed figure with its boundaries in the plane before we could arrive at the equivalent of the other assumption of Archimedes that ''of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least [in areal." This seems to be an impossible sense for l^ txrov even on. the assumption tnat it means "at an equal distance" in the present definition. The necessity therefore of interpreting li Sorov similarly in both definitions makes it impossible to regard it as referring to distance or length at all. It should be added that Simplicius gave the same explanations as Proclus (an-Nairld, p. 5). ' The language and construction of the definition. Let us now consider the actual wording and grammar of the phrase igfris ii WW TOic i^ Iavn7f mffjAioi^ Kcirai. As regards Sxe expression ii lo-ov we note that Plato and Aristotle (whose use of it seems typical^ commonly have it in the sense of "on a footing of equality": cf. 61 1( law m Plato's Laws 777 d, 919 d; Aristotle, Politics 1259 b 5 l{ lo-ov ctvat Povkerai r^v ^vcriv, "tend to be on an equality in nature," £tA, Nic. viii. 12, 1 161 a 8 IvravOa a-ovrcc i^ Iffw^ " there all are on a footing of equality." Slightly different are the uses in Aristotle, Eth. Nic. X. 8, 11 78 a 25 rcSv /icv y^ avayKotW xp«ta ical l{ Ztrov loTo^ "both need the necessaries of life to the same extent^ let us say"; Topics ix. 15, I74a32^ lirov woiovvra rrfv ipiDTtfiny, "asking the question indifferently" (i.e. without showing any expectation of one answer being given rather than another). The natural meaning would therefore appear to be "evenly placed" (or balanced), "in equal measure," "indifferently" or "without bias" one way or the other. Next, is the dative roU ^^* iavrq^ (n;/icioi$ constructed with i( urov or with Kfirai? In the first case the phrase must mean "that which lies evenly with (or in respect to) the points on it," in the second apparently "that which, in (or by) the points on it, lies (or is placed) evenly (or uniformly)." Max Simon takes the first construction to give the sense "die Gerade liegt in gleicher Weise wie ihre Punkte." If the last words mean " in the same way as (or in like manner as) its points," I cannot see that they tell us anything, although Simon attaches to the words the notion o( distance (Abstand) like Proclus. The second construction he takes as giving "die Gerade liegt fiir (durch) ihre Punkte gleichmassig," "the straight line lies symmetrically for (or through) its points"; or, if «c€irai is taken as the passive ofri^/u, "die Gerade ist durch ihre Punkte gleichmassig gegeben worden," "the straight line is symmetrically determined by its points." He adds that the idea is here direction, and that both direction and distance (as between two different given points simply) would be to Euclid, as later to Bolzano (Betrachtungen Oder einige Gegenstdnde der Elementargeometriey 1804, quoted by Schotten, Inhalt und Methode des planimetrischen Unterrichts, 11. p. 16), primary irreducible notions. While the language is thus seen to be hopelessly obscure, we can safely say that the sort of idea which Euclid wished to express was that of a line which presents the same shape at and relatively to all points on it, without any irr^ular or unsymmetrical feature distinguishing one part or side of it from another. Any such irr^ularity could, as Saccheri points out (Engel and Stackel, Die Theorie der ParalleUinien von Euklid bis Gauss, 1895^ p. 109X be at once made perceptible by keeping the ends fixed and turning the line about i68 BOOK I [ldbf. them right round; if any two positions were distinguishable, e.g. one being to the left or right relatively to another, "it would not lie in a uniform manner between its points." A conjecture as to its origin and meaning. The question arises, what was the origin of Euclid's definition, or, how was it suggested to him ? It seems to me that the basis of it was really Plato's definition of a straight line as "that line the middle of which covers the ends." Euclid was a Platonist, and what more natural than that he should have adopted Plato's definition in substance, while regarding it as essential to change the form of words in order to make it independent of any implied appeal to vision, which, as a physical fiict, could not properly find a place in a purely geometrical definition? I believe therefore that Eudid's definition is simply an attempt ^albeit unsuccessful, fix>m the nature of the case) to express, in terms to which a geometer could not object as not being part of geometrical subject-matter, the same thing as the Platonic definition. The truth is that Euclid was attempting the impossible. As Pfleiderer says (Scholia to Euclid), " It seems as though the notion of a straight Ime^ owing to its simplicity, cannot be explained by any regular definition which does not introduce words already containing m themsislv^ by implication, the notion to be defined (such eg. are duection, equali^, uniformity or evenness of position, unswerving course), and as though it were impossible^ if a person does not already know what the term siraigki here means, to teadi it to him unless by putting before him in some way a pictuie or a dtrawing of it" This is accordingly done in such books as Veronese's Eiemtnii H geofnetria (Part i., 1904, p. 10): ''A stretched string,* e.g. a plumniet, a ray of light entering by a small hole into a dark room, are rutiUmal objects. The image of them gives us the abstract idea of the limited line which is called a rectilineal segment!^ Other definitions. We will conclude this note with some other fan^ous definitions of a straight line. The following are given by Proclus (p. no, 18—23). I. A fine stretched to the utmost^ itr oKpov rcro/icn; ypa/A/Aif. This appears in Heron (ist c a.d.) also, with the words "towards the ends" (hri ra a^^ra) added. (Heron, ed. Hultsch, Def. 5, p. 8). a. Part of it cannot he in the assumed plane while part is in one higher up i^ fur€iap€T4pff). This is sl proposition in Euclid (xi. i). 3. jiU its parts Jit on all (other parts) alike, wavra avrfs rot fUfni www ofiolia^ iffMpfwiti, Heron has this too (Def. 5), but instead of "alike" he says iravTotm, "in all ways," which is better as indicating that the applied part may be applied one way or the reiferse way, with the same result 4. That line which^ when its ends remain fixed^ itself remains fixed^ ri rw w€paTW¥ fLcvoWttiv icttt aMi fiivovau. Heron's addition to this, " when it is, as it were^ turned round in the same plane*^ (olov h rf avnp lwvw&^ irrp€^ofUyti\ and his next variation, "and about the same ends having always the same position," show that the definition of a straight line as "that which does not change its position when it is turned about its extremities (or any two points in it) as poles" was no original discovery of Leibniz, or Saccheri, or Krafll, or Gauss, but goes back at least to the banning of the Christian era. Gauss' form of this definition was: "The line in which lie all points that, during the revolution of a body (a part of space) about two fixed points, maintain their position unchanged is called a straight line." SchoUen 7\ 1. DEFr. 4, S] NOTES ON DEFINITIONS 4. S 169 (i* P- 315) maintains that the notion of a straight line and its property of being determined by two points are unconsciously assumed in this definition, which is therefore a logical ''circle/' 5. Tkat line which with one other of the same species cannot complete a figure^ 17 /xrra r^s ^fiociSovs fuas o-x/ffAa firi dmrtkowra. This is an obvious SaT€poy'irp6T€poy, since it assumes the notion of a figure. Lastly Leibniz' definition should be mentioned: A straight line is one which divides a plane into two halves identical in all but position. Apart from the fact that this definition introduces the plane, it does not seem to have any advantages over the definition last but one referred to. Legendre uses the Archimedean property of a straight line as the shortest distance between two points. Van Swinden observes {Elemente der Geometries 1^341 P- 4)1 that to take this as the definition involves axjf#m//7^ the'proposition that any two sides of a triangle are greater than the third and proving that straight lines which have two points in common coincide throughout their length (cf- Legendre, Aliments de Geometric 1. 3; 8). The above definitions all illustrate the observation of Unger {Die Geometric des Euklidy 1833) : ^^ Straight is a simple notion, and hence all definitions of it must fail.... But if the proper idea of a straight line has once been grasped, it will b^ recognised in all the various definitions usually given of it ; all the definitions must therefore be regarded as explanations^ and among them that one is the best from which further inferences can immediately be drawn as to the essence of the straight line." Pefinition 5. A surface is that which has length and breadth only. The word ^irt^vcui was used by Euclid and later writers to denote surface in general, while they appropriated the word cViirffSov for plane surface, thus making htlw&w a species of the gentis iwif^tia, A sob'ts^ use of cn-i^oFcia by Euclid when a plane is meant (xi. Def. 1 1) is probably due to the fact that the particular definition came from an earlier textbook. Proclus (p. 116, 17) remarks that the older philosophers, including Plato and Aristotle, used the words lwi^v€ia and criircSov indifferently for any kind of surface. Aristotle does indeed use both words for a surface, with perhaps a tendency to use ciri^aycia more than criVcSov for a surface not plane. Cf. Categories 6, 5 a i sq., where both words are used in one sentence: "You can find a common boundary at wiiich the parts fit together»,a point in the case of a line, and a line in the case of a surface (cin^Vcia); for the parts of the surface (cViircSov) do fit together at some common boundary. Similarly also in the case of a body you can find a common boundary, a line or a surface (ciri^oi^cia), at which the parts of the body fit together." . Plato however does not use ciri^Vcca at all in the sense of surface, but only cv-iircSov for both surface and plane surfau. There is reason therefore for doubting the correctness of the notice in Diogenes Laertius, in. 24, that Plato "was the first philosopher to name, among extremities, ^'t plane surface " (cViircSos cv-c^ovcta). ffVi^cca of course means literally the feature of a body which is apparent to the eye (cirt^nfc), namely the surface. Aristotle tells us {De sensu 3, 439 a 31) that the Pythagoreans called a surface XP®^ which seems to have meant skin as well as colour, Aristotle explains the term with reference to colour (xp«i)f^) as a thing inseparable from the extremity (ir^«) of a body. X70 BOOK I Alternative definitions. The de6nitions of a surfiu» correspond to diose of a line. As in Aristotle a line is a magnitude ''(extended) one way, or in one 'dimension'** (c^* V^ "continuous one way" (c^' tr av¥€xk\ or *' divisible in one way" OMpaxJ Scoipcroi'), so a surface is a masnitude extended or continuous iwa ways (m Svo), or divisible in two ways (oixp). As in Euclid a surftce has "length and breadth '' only, so in Aristotle " breadth " is characteristic of the surface and is once used as synonymous with it (Meiafh. X020 a laX and again ''lengths are made up of long and short, surfaas of broad amd narrow^ and solids (<7km) of deep and shallow " (Meiaph. 1085 a 10). Anstotle mentions the common remark that a Une by Us moHon produces a surface {De anima i. 4, 409 a 4). He alsogives the a posteriori deKripti<Hi of a surface as the "extremitv of a solid** (Topics vi. 4, 141 b aa), and as "the section (rofiif) or division (fiuJpmn^) of a bo^" (ifetaph* 1060 b 14). Proclus remarks (p. 114^ ao) that we get a notion of a surfiure when we measure areas and mark their boundaries in the sense of length and breadth ; and we further get a sort of perception of it by looking at shadows, since these have no depth (for they do not penetrate the earth) but only have leqg^ and breadth. * - Classification of surfaces. Heron gives (Def. 75, p. 23, ed. Hultsch) two alternative divisions of surfaces into two classes, corresponding to Geminus* alternative divisions of lines, viz. into (i) incomposite and composite and (2) single and mixed. (i) Incomposite surfaces are ^ those which, when produced, fidl into (or coalesce with) themselves" (&rac UfioXkiiLwu a^roi tnJf iavni^ wArroiwtr), i.e. are of continuous curvature, e.g. the sphere. Composite surfaces are "those which, when produced, . cut one another." Of composite surfaces, again, some are (a) made up of non-homogeneous (elements) (c^ &,vo/ioiay€y£v) such as cones, cylinders and hemispheres, others lb) made up of homogeneous (elements), namely the rectilineal (or polyhedral) surfaces. (2^ Under the alternative division, simp/e surfaces are the plane and the sphencal surfaces, but no others ; the mixed class includes all other sur&ces whatever and is therefore infinite in variety. Heron specially mentions as belonging to the mixed class (a) the surfoce of cones, cylinders and the like, which are a mixture of plane and circular (fjLucraX ff^ cirtirffSov koI vcpt^cpcm) and (b) spiric surfaces, which are "a mixture of two circumferences " (by which he must mean a mixture of two drcular elements, namely the generating circle and its circular motion about an axis in . the same plane). Proclus adds the remark that, curiously enough, mixed surfaces may arise* by the revolution either of simple curves, e.^. in the case of the spire^ or of mixed curves, e.g. the "right-angled conoid" from a parabola, "another conoid" from the hyperbola, the "oblong" (c«>cffiyiccv, in Archimedes vapa- fioiccs) and " flat " (ImrAarv) spheroids from an ellipse according as it revolves . about the major or minor axis respectively (pp. 119, 6 — 120, 2). The hamoeo- meric surfaces, namely those any part of which wiU coincide with any other part, are two only (the plane and the spherical surface), not three as in the case of lines (p. 120, 7). J 1. DBFF. 6, 7] NOTES ON DEFINITIONS 5—7 171 Definition 6. *Eiri^aFffia« Sc wipara ypCLfjifAoL The extremities of a surface are lines. It being unscientific, as Aristotle says, to define a line as the extremity of a surface, Euclid avoids the error of defining the prior by means of the posterior in this way, and gives a different definition not open to this objection. Then, by way of compromise, and in order to show the connexion between a line and a surface, he adds the equivalent of the definition of a line previously current as an explanation. As in the corresponding Def. 3 above, he omits to add what is made dear by Aristotle {Metaph. 1060 b 15) that a "division" (Staipco-is) or "section" (ro/ii;) of a solid or body is also a surface, or that the common boundary at which two parts of. a solid fit together {Categories 6, 5 a 2) may be a surface. Proclus discusses how the fact stated in Def. 6 can be said to be true of surfaces like that of the sphere "which is bounded (ircirffpa<rraO, it is true, but not by lines." His explanation (p. 116, 8 — 14) is that "if we take the surface (of a sphere), so far as it is extended two ways (8ixig Scooran;), we shall find that it IS bounded by lines as to length and breadth ; and if we consider the spherical surface as possessing a form of its own and invested with a fresh quality, we must regard it as having fitted end on to beginning and made the two ends (or extremities) one, t^ing thus one potentially only, and not in actuality." ^ Definition 7. ^.^JVoW-^-- '^' ^ *" *EiriircSoc hrij^mftijL iarw, igfrif i( laov reus ^^* iavrrji cv^ciacf Kcirat. A plane surfieice is a surface which lies evenly with the straight lines on itself The Greek follows exactly the definition of a straight line mutatis mutandis^ i.e.' with raif...cvdffiaif for Tor«...cn7/xciOi«. Proclus remarks that, in general, all the definitions of a straight fa'ne can be adapted to the plane surface by merely changing the ^nus. Thus, for instance, a plane surface is " a surface the middle of which covers the ends " (this being the adaptation of Plato's definition of a straight line). Whether Plato actually gave this as the defini- tion of a plane surface or not, I believe that Euclid's definition of a plane surface as lying evenly with the straight litus on itself was intended simply to express the same idea without any implied appeal to vision (just as in the corresponding case of the definition of a straight line). I As already noted under Def. 4, Proclus tries to read into Euclid's defini- tion the Archimedeao assumption that "of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least" But, ' as I have stated, his interpretation of the words seems impossible, although it I is adopted by Simplidus also (see an-Nairizi). I Anjcient alternatives. The other ancient definitions recorded are as follows. I I. The surface which is stretched to the utmost {iw oKpov rcro/icn;) : a 1 definition which Proclus describes as equivalent to Euclid's definition ([on j Proclus' own view of that definition). Cf. Heron, Def. 11, "(a surface) which is right (and) stretched out " (opOi^ cSa-a dwoT€TafUinj\ words which he adds to Eudbid's definition. 1 73 BOOK I [l DBF. 7 2. 71k€ least surface among ail thast which have the same exiremiHet. Proclus is here (p. 1 17, 9) obviously quoting the Axchiaiedean assumfUom. 3. A surface all the parts of which have the property ofJUtmg am {each other) (Heron, Def. 11). 4. A surface such thai a straight Ume JUs on ail parts of it (Proclosi p. 117, 8), or such that the UraighiHne fits on iialiways^ i.e. however pboed (Proclus, p. 117, 20). With diis should be compared : 5. "(^ plane surface is) such thai^ if a straight line peus through two points on it, the line coincides wholly with it at every spot^ all ways/* ie. however placed (one way or the reverse, no matter how), 1^ ^rctJoy tvo ai^iMim Sifnim cvtfffui, jcol oktf avn; icara wearra rovor wamitK i^apfiiCenu, (Heron, Defl il). This appears, with the words mtra wivra riww wrolm omitted, in Theon of Smyrna (p. 112, 5, ed. HillerX so that it goes back at least as fiur as the ist c A.D. It is of course the same as the dfefinition qommonly attributed to Robert Simson, and very widely adopted as a substitute for Euclid's. This same definition appears also in an-Nairizi (ed. Curtse, p. zo) who, after quoting Simplicius' explanation (on the same lines as Proclus') of the meaning of Euclid's definition, goes on to say that ''others defined the plane surface as that in which it is possible to draw a straight line from any pdnt ! to any other." ♦ DiflBculties in ordinary definitions. Gauss observed in a letter to Bessel that the definition of a plane surfiioe as a surface such that^ if any two points in it ie tahen^ the straight line joining them lies wholly in the surface (which, for short, we will call ''Simson*^" definition) contains more than is necessary, in that a plane can be obtained by simply projecting a straight line lying in it from a point outside the line but also lying on the plane ; in fact the definition includes a theorem, or postulate, as well. The same is true of Euclid's definition of a plane as the surface which " lies evenly with (all) the straight lines on itself^" because it is suffident for a definition of a plane if the surface *' lies evenly " with those lines onl^ which pass through a fixed point on it and each of the several points of a straight line also lying m it but not passing through the point But from Euclid's point of view it is immaterial whether a definition contains more than the necessary minimum provided that the existence of a thing possessing all the attributes contained in the definition is afterwards proved. This however is not done in regard to the plane. No proposition about the nature of a plane as such appears before Book xi., .although its existence is presupposed in all the geometrical Books i. — iv. and vi. ; nor in Book xi. is there any attempt to prove, e.g. by construction, the existence of a surface conforming to the definition. The explanation may be that the existence of the plane as defined was deliberately assumed from the beginning like that of points and lines, the existence of which, according to Anstotle, must be assumed as principles unproved, while the existence of everything else must be proved ; and it may well«be that Aristotle would have included plane surfaces with points and lines in this statement had it not been that he generally took his illustrations horn plane geometry (excluding solid). But, whatever definition of a plane is taken, the evolution of its essential properties is extraordinarily difficult. Crelle, who wrote an elaborate article Zur Theorie der Ebene (read in the Academic der Wissenschaften in 1834) of which account must be taken in any full history of the subject, observes that, I. DEF. 7] NOTE ON DEFINITION 7 173 since the plane is the field, as it were, of almost all the rest of geometry, while a proper conception of it is necessary to enable Eucl. i. i to be understood, it might have been expected that the theory of the plane would have been the subject of at least the same amount of attention as, say, that of parallels. This however was far from being the case, perhaps because the subject of parallels (which, for the rest, presuppose the notion of a plane) is much easier than that of the plane. The nature of the difficulties as regards the plane have also been pointed out recently by Mr Frankland {The First Booh of Euclid's Elements^ Cambridge, 1905): it would appear that, whatever definition is taken, whether the simplest (as containing the minimum necessary to deter- mine a plane) or the more complex, e.g. Simson's, some postulate has to be assumed in addition before the fundamental properties, or the truth of the other definitions, can be established. Crelle notes the same thing as r^ards Simson's definition, containing more than is necessary. Suppose a plane in which lies the triangle ABC. Let AD join the vertex A ^ to any point D on BC^ and BE the vertex B to any J point E on CA. Then, according to the definition, AD lies wholly in the plane of the triangle; so does BE. But, if both AD and BE are to lie wholly in the one plane, AD^ BE must intersect, say at F\ if they did not, there would be two planes in question, not one. But the fact that the lines intersect and that, say, AD does not pass above or below BE^ is by no means self-evident. Mr Frankland points out the similar difficulty as regards the simpler definition of a plane as the surface generated by a straight line passing always through a fixed point and always intersecting a fixed straight line. Let OPF^ OQQ drawn from O intersect the straight line X 2X P^ Q respectively. Let R be any third point on X\ then it ^ needs to be proved that OR intersects P'Q in some ^ ^\ point, say R*. Without some postulate, however, it is not easy to see how to prove this, or even to prove that P'Q intersects X. Crelle's essay. Definitions by Fourier, Deahna, Becker. Crelle takes as the standard of a good definition that it shall be, not only as simple as possible, but also the best adapted for deducing, with the aid of the simplest possible principles, further properties belonging to the thing defined. He was much attracted by a very lucid definition, due, he says, to Fourier, according to which a platu is formed by the aggregate of ail the straight lines which^ passing through one point on a straight line in space^ are perpendicular to that straight line. (This is really no more than an adaptation from Euclid's proposition xi. 5, to the efiect tha^ if one of four concurrent straight lines be at right angles to each of the other three, those three are in one plane, which proposition is also used in Aristotle, Meteorologica in. 3, 373 a 13.) But Crelle confesses that he had not been able to deduce the necessary properties from this and had had to substitute the definition, already mentioned, of a plane as the surface containing^ throughout their whole lengthy all the straight lines passing through a fioud point and also intersecting a straight line in space \ and he only claims to have proved, after a long series of propositions, that the "Fourier"- or " perpendicular "-surface and the plane of the other definition just given ate identical, after which the properties of the " Fourier "-surface can be used along with those of the plane. The advantage of the Fourier definition is that it leads easily, by means of the two propositions that 174 BOOK I [l triangles are equal in all respects (i) when two sides and the induded an^ are respectively equal and (2) when all three sides are respectively equal, to the property expressed in Siroson's definition. But Crelle uses to establish these two congruence-theorems a number of propositions about equal angles^ si^k- meniary angles, right angles, greater and Use angles ; and it is difficult to question the soundness of Schotten's criticisin that these notions in themselves really presuppose that of a plane. The difficulty due to Fourier's use of the word '' perpendicular,*' if that were all, could no doubt be got over. Thus Deabna in a dissertation (Marburg, 1837) constructed a plane as follows. Presupposing the notions of a straight line and a qphere, he observes that, if a sphere revolve about a diameter, all the points of its surftce which move describe closed curves (circles). Each of mese circles, during the revolution, moves along itself, and one of them divides the sur&ce of the qpheie into two congruent parts. The aggrc^te then of the lines joining the centre to the points of this circle forms ih^ plane. Again, J. K. Becker (Die ElemetUe der Geometries 1877) pointed out that the revolution of a right ang^e about one side of it produces a conical suifice which differs from all other conical surfaces generated by the revolution of oAer angles in the foct that the particular cone coincides with the cone vertically opposite to ii : this characteristic might therefore be taken in order to get rid of the use of the rig^ angfe. W. Bolyai and Lobachewsky. Very similar to Deahna's equivalent for Fourier's definition is the device of W. Bolyai and LobachewBky (described by Frischauf, Elemente ier absoluten Geometries 1876). They worked upon a fundamental idea first suggested, apparently, by Leibniz. Briefly stated, their wi^ of evolving a plane and a straight line was as follows. Conceive an infinite number of pairs of concentric spheres described about two fixed points in space, (7, <7, as centres, and with equal radii, gradually increasing : these pairs of equal spherical surfaces intersect respectively in homogeneous curves (circles), and > the '* Inbegriff *' or aggr^ate of these curves of intersection forms a plane. If ^ be a point on one of these circles {k say), suppose points 3f, M' to start simultaneously from A and to move in opposite directions at the same speed till they meet at BsSBLy, B then is *' opposite" to A^ and A^ B divide the circumference into two equal halves. If the points ^, ^ be held fast and the whole system be turned about them until O takes the place of 0^ and O of Os the circle k will occupy the same position as before (though turned a different way). Two opposite points, P^ Q say, of each of the other circles will remain stationary during the motion as well as A, B: the " Inbegriff " or aggregate of all such points which remain stationary forms a straight line. It is next observed that the plane as defined can be generated by the revolution of the straight line about Off^ and this suggests the following construction for a plane. Let a circle as one of the curves of intersection of the pairs of spherical surfaces be divided as before into two equal halves at A^ B. Let the arc ADB be similarly bisected at Z>, and let C be the middle point of AB. This determines a straight line CD which is then defined as ''perpendicular" to AB. The revo- lution of CD about AB generates a plane. The property stated in Simson's definition is then proved by means of the congruence-theorems proved in Eucl. i. 8 and i. 4. The first is taken as proved, practically by considerations of symmetry and homogeneity. If two spherical surfaces, not necessarily equal, with centres O^ O intersHSCt, A and its ''opposite" point B are taken as \ L DEF. 7] NOTE ON DEFINITION 7 175 before on the curve of intersection (a circle) and, relatively to Off^ the point A is taken to be convertible with B or any other point on the homogeneous curve. The second (that of Eucl. i. 4) is established by simple application. Rausenberger objects to these proofs on the 'grounds that the first assumes that the two spherical surfaces intersect in one single curve, not in several, and that the second compares ang/es : a comparison which, he says, is possible only in a p/ant^ so that a plane is really presupposed. Perhaps as regards the particular comparison of angles Rausenberger is hypercritical; but it is difficult to r^ard the supposed proof of the theorem of Eucl. i. 8 as sufficiently rigorous (quite apart from the use of the uniform mo/ion of points for the purpose of bisecting lines). Simson's property is proved from the two congruence-theorems thus. Suppose that AB is *' perpendicular" (as defined by Bolyai) to two generators CJff CN of a plane, or suppose CM^ CN respectively to make with AB two angles congruent with one another. It is enough to prove that, if Z' be any pomt on the straight line MN^ then CP^ just as much as CM^ CW respectively, makes with AB two angles congruent with one another and is therefore a generator. We prove successively the congruence of the following pairs of triangles : ACM, BCM ACN, BCN AMN, BMN AMP, BMP ACP, BCP, whence the angles ACP, BCPzx^ congruent Other views. Enriques and Amaldi (EUmenti di geometria, Bologna, 1905), Veronese (in his EUmenti) and Hilbert all assume as a postulate the property stated in Simson's definition. But G. Ingrami {Elementi di geometria, Bologna, 1904) proves it in the course of a remarkable series of closely argued proposition oased upon a much less comprehensive postulate. He evolves the theory of the plane from that of a triangle, beginning with a triangle as a mere three-side (trilatero), i.e. a frame, as it were. His postulate relates to the three-side and is to the effect that each " (rectilineal) segment " joining a vertex to a point of the opposite side meets every segment similarly joining each of the other two vertices to the points of the sides opposite to them respectively, and, con- versely, if a point be taken on a segment joining a vertex to a point of the opposite side, and if a straight line be drawn from another vertex to the point on the segment so taken, it will if produced meet the opposite side. A triangle is then defined as the figure formed by the aggregate of all the segments joining the respective vertices of a thru-side to points on the opposite sides. After a series of propositions, Ingrami evolves a plane as the figure farmed by the " half straight-lines " which project from an internal point ^ the trian^e the points of the perimeter, and then, after two more theorems, proves that a plane is determined by any three of its points which are not in a straight line, and that a straight line which has tu^o points in a plane has all Us points in it. The argument by which Bolyai and Lobachewsky evolved the plane is of course equivalent to the definition of a plane as the locus ef all points eqmdistant from two fixed points in space. 176 BOOK I [l mrr. 7—9 Leibniz in a letter to Giordano defined a plane as ikai tmtfaa wUdk divides space into two congruemi parts. Admting to Giordano's cnticisni that you could conceive of surfaces and lines which divided qwce or a plane into two congruent parts without beipg/£iff^ or i/rw^A/ respectively. Bees (jOiar EuJUidische und NichtEuklidiseke Geameirie^ 1888) pointed out that what was wanted to complete the definition was the further condition that the two congruent spaces could be sHi tUong each other without the surfiuxs ceasiiy to coincide, and claimed priority for his completion of the definition in dus way. But the idea of all the parts of a i^ane fitting enctly on eUl other parts is ancient, appearing, as we have seen, in Heron, Det ii. Definitions 8, 9. 8. 'EvfircSoc Sc ywFui coTcv 1} i» hnwAf tuo jpa§t§aAy iano^ubm^ AXXajXmw fcoi fi^ iw* fMciac K€ifjL€Viiiiy wpit oAX^Xac tmk ypafi§im¥ kkun^ 9. ^Orav Sc al xi/Mcxovcnu r^ ytirior yfM^ngud Mumi Sau^, Mvypapt^un jcoXciTcu 17 yuyta. 8. A plane angle is the inelisuttion to one another of two Hnes m a pkme which meet one another and do mot lie in a straight Usie. 9. And when the lines containing the angit are straight^ the angle is tailed rectilineal. The phrase "not in a straight line** is strange, seeing that the definition purports to apply to angles formed by curves as well as straight lines. We should rather have expected eontmnous (om^xf^) with one another; and Heron takes this to be the meaning, since he at once adds an expknation as to what is meant by lines not being continuous (oA ovi^xm). It looks as though Euclid really intended to define a rectilineal angle, but on second thoughts^ as a concession to the then common recognition of curvilineal angles, altered ^ straight lines " into " lines " and separated the definition into two. I think ril our evidence suggests that Euclid's definition of an angle as inclination (xXuric) was a new departure. The word does not occur in Aristotle ; and we should gather fi-om him that the idea generallv associated with an angle in his time was rather deflection or breahing of lines (xXiurtv) : cf. his common use of #cc#cAairtfai and other parts of the verb fcXoK, and also his reference to one bent line forming an angle (r^r KtKOfjL^iiyqy koX ^^oiNroy yvn^iav^ Jfetaph, 1016 a 13). Proclus has a long and elaborate note on this definition, much of which (pp. 121, 12 — 126, 6) is apparently taken direct fi-om a work by his master Syrianus (6 iJ/«^cpos Ka$frf€fjMv). Two criticisms contained in the note need occasion no difficulty. One of these asks how, if an angle be an inclination, one inclination can produce two angles. The other (p. 128, 2) is to the effect that the definition seems to exclude an angle formed by one and the same curve with itself, e.g. the complete a'ssoid [at what we call the '^ cusp "1 or the curve known as the hippopede (horse-fetter) [shaped like a lemniscatej. But i such an "angle" as this belongs to higher geometry, which Euclid may well be excused for leaving out of account in any case. Other ancient definitions : ApoUonius, Plutarch, Carpus. Proclus* note records other definitions of great interest ApoUonius defined an angle as a contracting of a surface or a solid at one Ooint under a hrohen line or surface (onnayory^ crt^oyciav ^ oripcov 9po% l¥i oiffMcIy vwh K€KXaafUtqg ypo/^v ^ ^t^omigi), where again an angle is supposed to be formed by one broken line or surface. Still more interesting, perhaps, is the definition by "those who say that the first distance under the point (to vywrov I I. Dkff. 8, 9] NOTES ON DEFINITIONS 7—9 177 iiaarrifia vtrb ro ainUiov) is the angle. Among these is Plutarch, who insists that Apollonius meant the same thing ; for, he says, there must be some first distance under the breaking (or deflection^ of the including lines or surfaces, though, the distance under the point bemg continuous, it is impossible to obtain the actual firsty since every distance is divisible without limit " (cir* axc(f>ov). There is some vagueness in the use of the word " distance" {^^Mjtm\^ \ thus it was objected that ** if we anyhow separate off the/rj/" (distance being apparently the word understood) ''and draw a straight Ime through it^ we get a triangle and not one angle.'* In spite of the objection, I cannot but see in the idea of Plutarch and the others the germ of a valuable conception in infinitesimals, an attempt (though partial and imperfect) to get at the rate of divergence between the lines at their point of meeting as a measure of the angle between them. A third view of an angle was that of Carpus of Andoch, who said " that the angle was a quantity (itoctof), namely a distana {Stdarrifia) between the lines or surfaces containing it. This means that it would be a distance (or divergence) in one sense (c^* ty Sccorctfc), although the angle is not on that account a straight line. For it is not everything extended in one sense (to ^^' tv ScaoraroV) that is a line." This very phrase " extended one way" being held to define a line^ it is natural that Carpus' idea should have been described as the greatest possible paradox (Trarroii^ irapaSo^orarov). The difficulty seems to have been caused by the want of a different technical term to express a new idea ; for Carpus seems undoubtedly to have been anticipating the more modem idea of an angle as representing divergence rather than distance, and to have meant by c^' tv in one sense (rotationally) as distinct from one way or in one dimension (linearly). To what category does an angle belong? There was much debate among philosophers as to the particular category (according to the Aristotelian scheme) in which an angle should be placed ; is it, namely, a quantum (^fifT6v\ quale (iroiov) or relation (n-pos n) ? 1. Those who put it in the category o{ quantity argued from the fact that a plane angle is divided by a line and a solid angle by a surface. Since, then, it is a surface which is divided by a line, and a solid which is divided by a surface, they felt obliged to conclude that an angle is a surface or a solid, and therefore a magnitude. But homogeneous finite magnitudes, e.g. plane angles, must bear a ratio to one another, or one must be capable of being multiplied until it exceeds the other. This is, however, not the case with a rectilineal angle and the horn-like angle (iccparofi^'s), by which latter is meant the "angle" between a circle and a tangent to it, since (Eucl. in. 16) the latter "angle" is less than any rectilineal angle whatever. The objection, it will be observed, assumes that the two sorts of angles are homogeneous. Plutarch and Carpus are classed among those who, in one way or other, placed an angle among magnitudes-^ and, as above noted, Plutarch claimed Apollonius as a supporter of his view, although the word contraction \o{ a surface or solid) I used by the latter does not in itself suggest magnitude much more than Euclid's I inclination. It was this last consideration which doubtless led " Aganis," the ; " friend " (socius) apparently of Simplicius, to substitute for Apollonius' j wording " a quantity which has dimensions and the extremities of which arrive at one point'' (an-Nairizi, p. 13). 2. Eudemus the Peripatetic, who wrote a whole work on the angle, main- tained that it belonged to the category of quality. Aristotle had given as his fourth variety of quality "figure and the shape subsisting in each thincj. and, H. £. 12 I 17^ BOOK I [l Dcfp. 8, 9 besides these, straightness, curvature, and the like** (CMig^ries 8» 10 a 11). He says that each individual thing is spoken of as fuaie in respect of its fbmiy and he instances a triangle and a square, using them again later on (ifiU, 1 1 a 5) to show that it is not all qualities which are susceptible of man and less ; again, in Physics i. 5,, 188 a 25 a«^, strai^^ drcmlar are called kinds fA fy^re. Aristotle would no doubt have regarded deJUciwn (ffdcXdEirtfai) as belonging to the same category with straightness and curvature (fcofuraXoniv). At all eventi, Eudemus took up an angle as having its origin in the hnakmg or i^bdhn (fcAocrcc) of lines : deflection, he aigued, was quality if straightness was, and that which has its origin in quality is itself quality. Objectors to this view argued thus. If an an^le be a quality Qrmanyc) like heat or cold, how can it be bisCMSted, say ? It can m fact be divided ; and, if things of which divisibility is an essential attribute are varieties of quantum and not qualities, an angle cannot be a quality. Further, the man and the kss are the appropriate attributes tA quality, not the equal and the unequal ; if therefore an ande were a quality, we should have to say of angles, not that one is greater and another smaller, but that one is more an angle and another less an ang^ and that two angles are not unequal but dissimilar (nSrofUHOi). As a matter of fiurt, we are tokf by Simplicius, 538, 21, on Arist Da eaala that those who brought the an(^ uiKter the category of quale did call equal angles similar angles ; and Aristotle himself speaks of similar angles in this sense in Da coda 296 b 20, 311 b 34. 3. Euclid and all who cidled an angle an inclination are held by Sjrrianos to have classed it as a nlation (vpof rt). Yet Euclid certainly rqparded varies as magnitudes; this is clear both -from the earliest propositions deaSng specifically with angles, e.g. i. 9, 13, and also ^though in another way) from his describing an angle in the very next defimtion and always as camAumed (n€pi€xofuyri) by the two lines forming it (Simon, Euclid^ p. 28). Proclus (i.e. in this case Syrianus) adds that the truth lies between these three views. The angle partakes in fact of all those categories: it needs the quantity involved in magnitude, thereby becoming susceptible of equality, inequality and the like ; it needs the quality given it by its form^ and lastly the nlation subsisting between the lines or planes bounding it Ancient classification of '* angles." An elaborate classification of angles given by Proclus (pp. 126, 7 — 127, 16) may safely be attributed to (jeminus. In order to show it by a diagram it I Angles on sarfaces in solids I {iv OTcpewt) r- on simpU surfaces on mixed surfaces I (e.g. cones, cylinders) I ' 1 on phtus on spherical surfaces made by ntuple lines made by **mixid*^ lines bjr one of each (e.g. the angle made by a (e.g. the angle forAied by an curve, such as the cissM ellipse and its axis or by and kippapede, with itself) an ellipse and a drde) line-line line-drcumf. drcumf.-circumf. 1 1 1 . H ,, Ime-convcx Ime-concave convex-convex concave-concave ikiued, or (eg. an^e of a e.g. horn-like (dfu^vproi) {dft^oiKM) convex-concave semicircle) {xtparoti^t) or ••scraper-like" (e.g. those of ((MTporcdfcf) Utnes) 1. Deff. 8, 9] NOTE. ON DEFINITIONS 8, 9 179 will be necessary to make a convention about terms. Angles are to be under- stood under each class, " line-circumference *' means an angle contained by a straight line and an arc of a circle, " line-convex " an angle contained by a straight line and a circular arc with convexity outwards^ and so on in every case. Definitions of angle classified. As for the point, straight line, and plane, so for the angU^ Schotten gives a valuable summary, classification and criticism of the different modem views up to date (Inhalt und Methode des planimetrischen Unterrichts^ 11., 1893, pp. 94 — 183); and for later developments represented by Veronese reference I may be made to the second article (by Amaldi) in Questioni riguardanti \ la geometria eUmentare (Bologna, 1900) already referred to. I With one or two exceptions, says Schotten, the definitions of an angle may f be classed in three groups representing generally the following views : \ ' I. The angle is the difference of direction between tUH> straight lines. (With JL this group may be compared Euclid's definition of an angle as an inclination.) I 2. The angle is the quantity or amount (or the measure) of the rotation \ necessary to bring one of its sides from its own position to that of the other side without its moving out of the plane containing both, 3. The angle is the 'portion of a plane included between ttvo straight lines in the plane which meet in a point {or two rays issuing from the point). It is remarkable however that nearly all of the text-books which give definitions different from those in group 2 add to them something pointing to a connexion between an angle and rotation : a striking indication that the essential nature of an angle is closely connected with rotation, and that a good definition must take account of that connexion. The definitions in the first group must be admitted to be tautologous, or circular^ inasmuch as they really presuppose some conception of an angle. Direction (as between two given points) may no doubt be regarded as a primary k notion; and it may be defined as "the immediate relation of two points which } the ray enables us to realise'' (Schotten). But ''a direction is no intensive ^ magnitude, and therefore two directions cannot have any quantitative I difference" (Biirklen). Nor is direction susceptible of differences such as 'those between qualities, e.g. colours. Direction is a jiVi^ar' entity: there cannot be different sorts or degrees of direction. If we speak of "a different direction," we use the word equivocally ; what we mean is simply "another" direction. The fact is that these definitions of an angle as a difference of direction unconsciously appeal to something outside the notion of direction . altogether, to some conception equivalent to that of the angle itself. Recent Italian views. I The second group of definitions are (says Amaldi) based on the idea of the rotation of a straight line or ray in a plane about a point : an idea which, I logically formulated, may lead to a convenient method of introducing the I angle. But it must be made independent of metric conceptions, or of the conception of congruence^ so as to bring out first the notion of an angle, and afterwards the notion of ^^wa/ angles. The third group of definitions satisfy the condition of not including metric conceptions ; but they do not entirely correspond to our intuitive conception of an angle, to which we attribute the character of an entity in one dimension i(as Veronese says) with respect to the ray as element, or an entity in tivo 12- iSo BOOK I [l Dsr. 9 dimensions with reference Xo points as elementSi which may be called an attgtdar sector. The defect is however easily remedied by considering the an^ as " the aggr^te of the rays issuing from the vertex and comprised in the angular . sector." Proceeding to consider the principal methods of arriving at the logical formulation of the first superficial properties of the ftam from whidi a definition of the angle may emerge, Amaldi distinguishes two points of view (1) XhtgtfutiCy (2) the actual. (i) From the first point of view we consider the dmster of straight Mnes or rays (the aggregate of all the straight lines in a plane oassing throush a point, or of all the rays with their extremities in that point) as generatea by the movement of a straight line or ray in the plane, about a point This leads to the postulation of a closed order^ or circular disposition^ of the straight lines or rays in a cluster. Next comes the connodon subsisting between the disposition of any two clusters whatever in one plane, and so oa (2) Starting from the point of view of the aitual^ we lay the foundation of the definition of an angle in the dimmn of the plane into two parts pialf- planes) by the straight line. Next, two straight lines (tf, i) in the plane, inter- secting at a point O^ divide the plane into four regions which are called angular sectors (convex) ; and finally the an^e (ab) or (Ai) may be defined as the aggregate of the rays issuing firam O CMd belonging to the angular sector which has a and hfor sides. Veronese's procedure (in his Elementi) is as follows. He bq;ins with the first properties of the plane introduced by^ the following definition. The figure given by all l^e straight lines joining Uie pomts of a straight line r to a point /' outside it and by the parallel to r through /' is called a cluster of straight lineSy a cluster of rays^ or a plane^ according as we consider the element of the figure itself to be the ^^y"^ j straight line^ the ray terminated at F^ P ^ or z, point. I [It will be observed that this method of producing a plane involves using { the parallel to r. This presents no difficulty to Veronese because he has 4 previously defined parallels, without reference to the plane, by means of reflex or opposite figures, with respect to a point O : '' two straight lines are csdled < parallel^ if one of them contains two points opposite to (or the reflex of) two points of the other with respect to the middle point of a common transversal (of the two lines)." He proves by means of a postulate that the parallel r does belong to the plane Pr. Ingrami avoids, the use of the parallel by defining a plane as "the figure formed by the half straight lines which project ( 'fi-om an internal point of a triangle (i.e. a point on a line joining any vertex of a three-side to a point of the opposite side) the points of its perimeter," and " then defining a cluster of rays as "the aggregate of the half straight lines in a plane ^starting from a given point of the plane and passing through the points I of the perimeter of a triangle containing the point."] J Veronese goes on to the definition of an angle. " We call an angle apart of a duster o/rays^ bounded by two rays (as the segment is a part of a straight ' line bounded by two points). ^*An angle of the cluster^ the bounding rays of which are opposite^ is caUod a flat angle." Then, after a postulate corresponding to postulates which he lays down for I. Deff. 9-12] NOTES ON DEFINITIONS 9—12 181 a rectilineal segment and for a straight line^ Veronese proves that all fiat angles are equal to one another. Uf9 Hence he concludes that "the duster of rays is a homogeneous linear system in which the element is the ray instead of the point. The cluster being a homogeneous linear system, all the propositions deduced from [Veronese's] Post i for the straight line apply to it, e.g. that relative to the sum and difference of the segments : it is only necessary to substitute the ray for the point, and the angle for the segment.*' Definitions 10, 11, 12. 10. *Orav 8c cMciOi lit cMcIaF orrotfcura ras ^^€^79 ytavta.% uras liXXi/Xaiv ir<k|J, ^ptfi; kKuripa toiv Tcraiv fa¥%mv inrif koI 17 ^^con^jcvia cMcui icd^croc fcoAcirai, 1 1. *Afi)8Xcia yiavia iariv 1} /ittltop 6pBrj^ 12. *Ofcui Sk ff IKaaatav 6p$7Js. 10. fVhen a straight line set up on a straight line makes the adjacent angles equal to one anothery each of the equal angles is right, and the straight line stcmding on the other is called a perpendicular to that on which it stands, I II. An obtuse angle is an angle greater than a right angle. \ 1 2. An acute angle is an angle less than a right angle, \ ^^€^ is the regular term for adjacent angles, meaning literally " (next) in I ' order.'' . I do not find the term used in Aristotle of angles^ but he explains its i meaning in such passages as Physics vi. i, 231 b 8 : "those things are (next) i in order which have nothing of the same kind {crvyytvk) between them." ► fca^crof , perpendicular^ means literally let fall : the full expression is perpen- I dicular straight line, as we see from the enunciation of Eucl. i. 11, and the \ notion is that of a straight line let fall upon the surface of the earthy dLpiumb- [ line, Proclus (p. 283, 9) tells us that in ancient times the perpendicular was called gnomon-tvise (#cara yv«ifioFa), because the gnomon (an upright stick) wtt^ f set up at right angles to the horizon. The three kinds of angles are among the things which according to the Platonic Socrates {Republic vi. 510 c) the geometer assumes and argues from, declining to give any account of them because they are obvious. Aristotle discusses the priority of the right angle in comparison with the acute (Metaph. 1084 b 7): in one way the right angle is prior, i.e. in being defined (^n Zpurraj) and by its notion (rf Xoy<{»), in another way the acute is prior, i.e. as I being a party and because the right angle is divided into acute angles ; the acute angle is prior as mattery the right angle in respect oi form\ cf. also Metaph, 1035 b 6, ''the notion of the right angle is not divided into Definition 14. 182 BOOK I [i. Deff. 12-14 that of an acute angle, but the reverse ; for, when defining an acute angle, you make use of the right angle.*' Proclus (p. 133, 15) observes that it is by the perpendicular that we measure the heights of figures, and that it is by reference to the right angle that we distinguish the other rectilineal angles, which are otherwise undistinguished the one from the other. The Aristotelian Problems (16, 4, 913 b ^6) contain an ex|xression perhaps worth quoting. The question discussed is why things which &11 on the ground and rebound mikt ''similar" an^es with thMuraice on both sides of | the point of impact; and it is observed that '^.die right angle is the Nmii ^ (ofMs) of the opposite angles," where however *' opposite " seems to mean, not ''supplementary " (or acute and obtuse), but the equal angles made with the ^ surface on opposite sides of the perpendicular. Proclus, after his manner, remarks that the statement that an angle less than a right angle is acute is not true without qualification, for f i]| the kmihUke angle (between the circumference of a drde ai\d a tangent) is less than a right angle, since it is less than an atuU angle, but is not an acute angle, wbik (2) the "angle of a semicircle " (between the arc and a diameter) is also less than a right angle,* but is not an acute angle. % The existence of the right angle is X)f course proved in L 1 1. Definition 13. Opo% iariv, o tivos iari wipa$. A boundary is that which is an extremity rfemy thing. Aristotle also uses the words 2pof and t^s as synonymous. Cf. Drgm, animaL ii. 6, 745 a 6, 9, where in the expression "h'mit of magnitude** first one and then the other word is used. Proclus (p. 136, 8) remarks that the word boundary is appropriate to the origin of geometry, which began from the measurement of areas of ground and involved the marking of .boundaries. i Sx^fui loTi TO viro TIV09 17 rivwv op«iiv mpuxofityov, . A figure is that rvhich is contained by any boundary or boundaries. \ Plato in the Meno observes that roundness (trrpor^Xifnfi) or the round is a ^ " figufe," and that the straight and many other thmgs are so too ; he then j inquires what there is common to all of thejxi, in virtue of which we apply the | term "figure" to them. His answer is (76 a): "with reference to every j fi^re I say that that in "u^hich the solid terminates (rovro^ ci9 u rh oryMlw irtfHuv€C) is a figure^ or, to put it briefly, a figure is an extremity of a solid.** I The first observation is similar to Aristotle's in the Physics i. 5, 188 a 25, < where angle, straight, and circular are mentioned as genera of figure: In the Categories 8, 10 a 11, "figure" is placed with straighthess and curvedness in the category' of quality. Here however "figure" appears to mean skafe I (fiap^') rather than " figure " in oiir sense. Coming hearer to "figure" in our ! sense, Aristotle admits that figure is ""a sort of magnitude" i^De anima in. i, \ 425 a 18X and he distinguishes plcme figures of two kinds, in language not unlike Euclid's, as contained by straight and circular lines re^)ectively : "every plane figure is either rectilined or formed by circular- lines (vcpt^cpoypa/ifuiy), and the rectilineal figure is contained by several lines, the circular by one line " {De caelo 11. 4, 286 b 13). He is careful to explain that a plane is not a I. Deff. 14-16] NOTES ON DEFINITIONS 12—16 183 figure, nor a figure a plane, but that a plane figure constitutes one notion and is a species of the genus figure {AnaL post, n. 3, 90 b 37). Aristotle does not attempt to define figure in general, in fact he says it would be useless : " From this it is clear that there is one definition of soul in the same way as there is one definition oi figure \ for in the one case there is no figure except the triangle, quadrilateral, and so on, nor is there any soul other than those above mentioned. A definition might be constructed which should apply to all figures but not specially to any particular figure, and similarly with the species of soul referred to. [But such a general definition would serve no purpose.1 Hence it is absurd here as elsewhere to seek a general definition which will not be properly a definition of anything in existence and will not be applicable to the particular irreducible species before us, to the neglect of the definition which is so applicable" {De anima 11. 3, 414 b 20^ — 28). Comparing Euclid's definition with the above, we observe that by. intro- ducing boundary {opos) he. at once excludes the straight which Aristotle classed as figure ; he doubtless excluded angle also, as we may judge by (i) Heron's statement that ''neither one nor two straight lines can complete a figure," (3) the alternative definition of a straight line as *'that which cannot with another line of the same species form a figure," (3) Geminus' distinction between the line irhxcYi forms a figure (crxfffJLarowoiovaa) and the line which extends indefinitely {iw* aTrcipov iKpaXXxt/uvri), which latter term includes a hjrperbola and a parabola. Instead of calling figure an extremity as Plato did in the expression '^extremity (or limit) of a solid,** Euclid describes a figure as that which has a boundary or boundaries. And lastly, in spite of Aristotle's objection, ht does attempt a general definition to cover all kinds of figure, solid and plane. It appears certain therefore that Euclid's definition is entirely his own. Another view of a figure, recalling that of Plato in Meno 76 a, is attributed by Proclus (p. 143, 8) to Posidonius. The latter regarded the.^^r^ as the confining extremity or limit (wipas cnrvicXciov), "separating the notion of figure firom quantity (or magnitude) and making it the cause of definition^ limitation^ and inclusion (roO iipL<r$a%, icoi ir€W€pcur6<u ical t^ itc^mox^s)... Posidonius thus seems to have in view only the boundary placed round from outside, Euclid the whole content, so that Euclid will speak of the circle as a figure in respect of its whole plane (surface) and of its inclusion (firom) without, whereas Posidonius (makes it a figure) in respect of its circumference... Posidpnius wished to explain the notion of figure as itself limiting and confining magnitude." Proclus observes that a logical and refining critic might object to Euclid's definition as defining the genus from the species, since that which is enclosed by one boundary and that which * is enclosed by several are both species of figure. The best answer to this seems to be supplied by the passage of ^ Aristotle's I>e anima quoted above. ; Definitions ,15, 16. ^ I 15. KvfcXos iari <r}(rjfta ciriircSov vwo fiuis ypofifi^c mpuxofifvoy [^ fcaXcinu T^H^M^>cta|, wp^ ^¥ d^* M^ fnffJL€iov rwv cvros tov fr^mpxiro^ icccficvaiv n-curai at \ wpovwiirrownii cii^cuu {irpof rrjv rov kvkXov ir€pt/^€p€iay\ urcu dXAifXacc curiv. ' 1 6. Kivrpw 8i rov.KVicXov ro airfp,€iO¥ fcaXcirac 1$. A circle is a J>lane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another ; 16. And the point is called the ctnXxe of the circle. i84 BOOK I [i. Dbff. t5» i6 The words ^ KoXctTai xcpc^^io, *' which is called the drcumferenoe," and wpoi rifv Tov kvkXov w€piitiip€ua»f "to the circuinference of the drde," aie bracketed by Heiberg because, although the Mss. have them, they aie omitted in other ancient sources, viz. Prmdus, Taurus, Sextus Empiricus and Boethius, and Heron also omits the second gloss. The recently discovered papyrus Herculanensis No. 1061 also quotes the definition without the words m question, confirming Heibeig's rejection of them (see Heiberg in Hermes XXXVIII., 1903, p. 47). The words were doubtless added in view of the occurrence of the word "circumference'' in Deff. 17, 18 immediatdv following, without any explanation. But no explanation was needed. Though the word xc/M^cpcia does not occur in Plato, Aristotle uses it seyeml timet ( 1 ) in the general sense of contour without any special mathematical signification, (2) mathematically, with reference to the rainbow and the circumference, as well as an arc, of a circle. Hence Euclid was perfectly justified in emtdoying the word in Deff. 17, 18 and elsewhere, but leaiong it undefined as being a word universally understood and not involving in itself any mathematical conception. It may be added that an-NairlzI had not the bracketed words in his text ; for he comments on and tries to explain Euclid's omission to define the circumference. The definition itself contained nothing new in substance. Plato {Parme- nides 137 e) says : " Round is, I take it, that the extremes of which are every way equally distant from the middle " (irrpoyyvAor yc vov Icrrt ToOr«s oS & td laxora warraxj awo rav ficcrov lirov Awixff). In Aristotle we find the following expressions: "the circular (xfpi^€pdypo/4fu>v) plane figure bounded by one line" (Df caelo 11. 4, 286 b 13 — 16); "the plane equal (Le. extending equally all ways) from the middle" (IrArcSoK to lie rov /mctov \ffw\ meaning a circle (^Rhetoric iii. 6, 1407 b 37); he also contrasts with the circle ^'any other figure which has not the lines from the middle equal, as for example an egg-shaped figure" {De caelo ii. 4, 287 a 19). The word "centre" («€»^pw) was also regularly used : cf. Proclus* quotation from the "oracles" (Aoyia), " the centre from which all (lines extending) as far as the rim are equal." The definition as it stands has no genetic character. It says nothing as to the existence or non-existence of the thing defined or as to the method of constructing it. It simply explains what is meant by the word " circle," and is a provisional definition which cannot be used until the existence of circles is proved or assumed. Generally, in such a case, existence is proved by actual construction; but here the possibility of constructing the circle as defined, and consequently its existence, are /^j/^/17/^// (Postulate 3). A genetic definition might state that a circle is the figure described when a straight line, always remaining in one plane, moves about one extremity as a fixed point until it returns to its first position (so Heron, Def. 29). Simplicius indeed, who points out that the distance between the feet of a pair of compasses is a straight line from the centre to the circumference, will have it that Euclid intended by this definition to show how to construct a circle by the revolution of a straight line about one end as centre ; and an- Nairizi points to this as the explanation (i) of Euclid's definition of a circle as 2i plane figure^ meaning the whole surface bounded by the circumference, and not the circumference itself, and (2) of his omission to mention the "circumference," since with this construcrion the circumference is not drawn separately as a line. But it is not necessary to suppose that Eudid himself did more than follow the traditional view ; for the same conception of the circle as k plane figure appears, as we have seen, in Aristotle. While, however. If I. Deff. 15-17] NOTES ON DEFINITIONS 15—17 185 Euclid is generally careful to say the ^^circumference of a circle " when he means the circumference, or an arc, only, there are cases where "circle" means "circumference of a circle," e.g. in in. 10: "A circle does not cut a circle in more points than two." Heron, Produs and Simplidus are all careful to point out that the centre is not the only point which is equidistant from all points of the circumference. The centre is the only point in the plane of the circle ("lying within the figure," as Euclid says) of which this is true; any point not in the same plane which is equidistant from all points of the circumference is a pote. If you set up a "gnomon " (an upright stick) at the centre of a circle (i.e. a line through the centre perpendicular to the plane of the circle), its upper extremity is a pole (Proclus, p. 153, 3); the perpendicular is the locus of all such poles. Definition 17. Ataficr/>09 8c rov kvkXov ktrrXv cMcui ric 81a rw Kiirrpov rjyitiyti ical irtparov- ficn; i^* hcaripa ra /upij vro r^^ rev kvkXov w€pif^€p€ias, i/ric icoi &\a tc/avci rov JCVfcAoF. A diameter of the circle is any straight lifie drawn through the centre and terminated in both directions by the circumference of the circle^ and such a ^reught line also bisects the circle. The last words, literally "which (straight line) also bisects the circle," are omitted by Simson and the editors who followed him. But they are necessary even though they do not "belong to the definition" but only express a property of the diameter as defined. For, without this explanation, Euclid would not have been justified in describing as a j^/rrAnrcle a portion of a circle bounded by a diameter and the circumference cut off by it Simplicius observes that the diameter is so called because it passes through the whole surface of a circle as if measuring it, and also because it divides the circle into two equal paits. He might however have added that, in general, it is a line passing through a figure where it is widest^ as well as dividing it equally: thus in Aristotle ra #cara huL/urpoy icct/Acva, "things diametrically situated ** in space, are at their maximum distance apart Diameter was the regular word in Euclid and elsewhere for the diameter of a square^ and also of a parallelogram; diagonal (Siayuvios) was a later term, defined by Heron (Def. 68) as the straight line drawn from an angle to an angle. Proclus (p. 157, 10) says that Thales was the first to prove that a circle is bisected by its diameter; but we are not told how he proved it Proclus gives as the reason of the property " the undeviating course of the straight line through the centre " (a simple appeal to symmetry), but adds that, if it is desired to prove it mathematically, it is only necessary to imagine the diameter drawn and one part of the circle applied to the other ; it is then clear that they must coincide, for, if they did not, and one fell inside or outside the other, the straight lines from the centre to the circumference would not all be equal : which is absurd. Saccheri's proof is worth quoting. It depends on three "Lemmas" immediately preceding, (i) that two straight lines cannot enclose a space, !2^ that two straight lines cannot have one and the same segment common, 3) that, if two straight lines meet at a point, they do not touch, but cut one another, at it. " Let MDHNKM be a circle, A its centre, MN a diameter. Suppose i i86 BOOK I [i. Dkff. 17, the portion MNKM of the circle turned about the fixed points M^ N^ 90 that it ultimately comes near to or coincides with the remaining portion MNHDM. "Then (i) the whole diameter MAl^^ with all its points, clearly remains in the same position, since otherwise two straight lines would enclose a space (contrary to the first Lemma). '' (ii) Clearly no point K of the circumference NKM falls withm or outside the surfieure enclosed by the diameter if^A^and the other part, JV»Z>i/; v \/ J of the circumference, since otherwise, contrary to N. ^x M the nature of the circle, a radius as AK would be ^"^ --^ I less or ^eater than another radius as AH. \ *' (iii) Any radius MA can clearly be rectilineally produced only along a single other radius AN^ since otherwise (contrary to the second Lemma) two lines assumed straight, e.g. MAN^ MAH^ would have one and the same ^ common segment " (iv) All diameters of the circle obviously cut one another in the centre (Lemma 3 preceding), and they bisect one another there, by the go^eral properties of the circle. " From all this it is manifest that the diameter MAN divides its circle and the circumference of it just exactly into two equal parts, and the same ma^ be generally asserted for every diameter whatsoever of the same circle ; which was to be proved." Simson observes that the property is easily deduced from in. 31 and 24 ; for it follows from iii. 31 that toe two parts of the circle are ^similar segments" of a circle (segments containing equal angles, in. Def. 11), and from III. 24 that they are equal to one another. Definition 18. 'HfUJcvicXuiy Sc Itm ro irtpvf^iktvov vyri^a. ^ro re rSjs Sta/icrpov fcot r^ , dxoAafi)3ako/Mn^ vtt* aMj^ ir€pi/^€p€ia%, Kivrpov 8c rov iffiucvKkiov r^ at^ i I icoi rov kvkXov IcrrtV. y A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. The last words, '*And the centre of the semicircle is the same as that ( of the circle," are added from Proclus to the definition as it appears in the Mss. Scarburgh remarks that a semicircle has no centre, properly speaking, | and thinks that the words are not Euclid's, but only a note by Proclus. I am \ however inclined to think that they are genuine, if only because of the very * futility of an observation added by Proclus. He explains, namely, that the ' semicircle is the only plane figure that has its centre on its perimeter (!), "so that you may conclude that the centre has three positions, since it may be i within the figure, as in the case of a circle, or on the perimeter, as with the I semicircle, or outside, as with some conic Unes (the single-branch hyperbola ' presumably)" ! > Proclus and Simplicius point out that, in the order adopted by Euclid for these definitions of figures, the first figure taken is that bounded by one line (the circle), then follows that bounded by two lines (the semicircle), then the triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes 1 :v 1. Deff. 18-21] NOTES ON DEFINITIONS 17—21 187 different kinds of figures bounded by two lines (pp. 159, 14 — 160, 9). Thus they may be formed (i) by circumference and circumference, e.g. (a) those forming angles, as a /un^ (to fArfvo€t&€^) and the figure included by two arcs with convexities outward, and (d) the angle-less (dycJi^iOF), as the figure included between two concentric circles (the coronal) \ (2) by circumference and straight line, e.g. the semicircle or segments of circles (a^iScc is a name given to those less than a semicircle); (3) by "mixed" line and "mixed" line, e.g. two ellipses cutting one another; (4) by " mixed " line and circumference, e.g. intersecting ellipse and circle ; (5) by " mixed " line and straight line, e.g. half an ellipse. Following Def. 18 in the mss. is a definition of a segment of a circle which was obviously interpolated from 111. Def. 6. Proclus, Martianus Capella and Boethius do not give it in this place, and it is therefore properly omitted. Definitions 19, 20, 21. 19. Sx^fuira tifOvypafAfia ioTi rk vro cii^cuSv ir^icxoficva, rpiVXcvpa /acv ra viro rprnVf rcrpairXcvpa Sc ra {fwo rc<r<rapci>v, woKvvkwpa 8c ra vro nk^iovtav ij Ttavapiav €vOtiwv ir€pi€XPfi€V€L 20. Toiv &k rpivkwpmv <ryyiimwv Urivktvpov fiky Tpiyuiv6y iari ro ra« rpcts urac ixpv xXcvpac, uroa'K€kU Sk to rac fivo fi6yas Icras Ix^vk vXcvpav, (ffcaXiTvov Sk TO Ttts Tpcis dvurov^ l)(ov wktvpas, -^ 21. *ETi Sc TOIV TpiirXcvp<Dv <r)(iffMariav dpOaytivtov /acv rplywyop Icrri to c^ov 6fArjv ywvlavy dfiPkvyiaviov 8c to I^of d/ifiktZay ycDFtiav, 6$vyiivtoy 8c to Tas TpciS 19. Rectilineal figures are those which are contained by straight lines^ trilateral figures being those contained by three^ quadrilateral those contained by four^ and multilateral those contained by more than four straight lines, 20. Of trilateral figures, an equilateral triangle fx that which has its three sides equals an isosceles triangle that which has two of its sides alone equcU^ and a scalene triangle thcU which has its three sides unequal. 2 1. Further^ of trilateral figures^ a right-angled triangle is that which has a right angle^ an obtuse-angled triangle that which has an obtuse angle^ and an acute-angled triangle that which has its three angles acute, 19. The latter part of this definition, distinguishing three-sided^ four-sided and many-sided figures, is probably due to Euclid himself, since the words TpiirXcvpov, TCTpan-Xcvpov and xoXvVXcvpok do not appear in Plato or Aristotle (only in one passage of the Mechanics and of the Problems respectively does even rcTpaVXcupov, quadrilateral^ occur). By his use of TCTpaVXcvpov, quadrilateral, Euclid seems practically to have put an end to any ambiguity in the use by mathematicians of the word TCTpayuvov, literally "four-angled (figure)," and to have got it restricted to the square. Cf. note on Def. 22. 20. Isosceles {uroaictkiii, with equal legs) is used by Plato as well as Aristotle. Scalene (<rKakr/y6% with the varient frKakrpnj^) is used by Aristotle of a triangle with no two sides equal: cf. also Tim. Locr. 98 b. Plato, Euthyphro 12 d. \ i88 BOOK I [t DiPP. so, sa applies the term '' scalene " to an odd number in contrast to ^ isosceles " used of an even number. Proclus (p. i68, 24) seems to connect it with 9taiff^ to limp ; others make it akin to irmAi^ aroohedy aslani. Apollonius uses the same word " scalene '* of an obUpti circular cone. Triangles are classified, first with reference to their sides, and then with reference to their angles. Produs points out that seven distinct species of triangles emerge: (i) the equiiiUirai triangle, (a) three species of isasteies triangles, the r^;ht-angled, the obtuse-angM and the acute-angled, (3) the same three' varieties of scalene triangles. Proclus gives an odd reason for the dual classification according to sides and angles, namely that Euclid was mindful of the ftct that it is not every triangle that is trilateral alsa He explains this statement by reference (p. 165, 22) to a figure which some called barb4iki (d«ctotiSiyv) while Zenodorus odled it koUauhangkd (MocXoyvJMoc). Proclus mentions it again in his note on i. 22 (p. 328, 21 sqq[.) as one 61 the paradoxes of f;eoinetr]|r, observing that it is seen in the figure of that proposition. This "triangle" is merely a quadrilateral with a re-entrant angle ; and the idea that it has only three angles is due to the n(m-recognition of the fourth angle (which is greater than two right angles) as being an angle at all. Since Proclus speaks of \h<t fomr-sided trian^ as ''one of the paradoxes in geometry,'' it is perhaps not sdfe to assume that the misconception underlying tl^ expression existed in the mind of Proclus alone ; but there does not seem to be any evidence that Zenodorus called the figure in question a triangle (cf. Pappus, ed. Hultsch, pp. 1 1 54, 1206). I Definition 23. j Tcav 8i rcrpaxXcvpuiv axflfdrtav rtrpaywvov fiiv loriv, i UrAwXoffi^w ri ion I fcoi 6p0oywtoVf Ircpofii^fccs &c^ i 6p6oyw¥iov fih^^ oIk uFoirXcvpov hiy ^fLpo9 8c, i I laowXwfiw /livf cWmc 6p6oyw¥iO¥ 8^ (&o/aJSoci8<s Sk ro r&s Airtyavrtov wkwvpai re irai i ytiwia,^ uras dXAi^Acus ^XP^i ^ ^^ laowXtvpov Itrrw ovt€ 6pOoyiayioy ra 8i wapa ravra rerpdrrXtvpa rpa7rc(ia KaXturdm, I Of quadrilateral figures^ a square is that which is both equilateral and right- k angled; an oblong that which is right-angled but not equilateral; a rhombus \ that which is equilateral but not right-an^d; and a rhomboid that which has ^ its opposite sides and angles equal to one another but is neither equilateral nor \ right-angUd. And let quadrilaterals other than these be called trapezia. . rcrfMywrov was already a square with the Pythagoreans (cf. Aristotle, Metaph, 986 a 26), and it is so most commonly in Aristotle ; but in De anima < II. 3, 414 b 31 it seems to be a quadraateral, and in Metaph, 1054 b 2, ( '' equal and equiangular rcrpayoiva," it cannot be anything else but quadri- ' lateral if "equiangular" is to have any sense. Though, by introducing { TtrpanrXitvpw for any quadrilateral, Euclid enabled ambiguity to be avoided, there seem to be traces of the older vague use of rcrpayuiov in much later writers. Thus Heron (Def. 104) speaks of a cube as "contained by six equi- I lateral and equiangular r^rpaytava" and Proclus (p. 166, 10) adds to his ! remark about the " four-sided triangle ** that " you might have ra-paytava with more than the four sides," where TtTpdytova can hardly mean squares. ' Ircpdfii^iccf, oblong (with sides of different length)^ is also a Pythagorean term. The word right-angled (^oywiov) as here applied to quadrilaterals must mean rectangular (Le., practically, having all its angles right angles); for, although it is tempting to take the word in the same sense for a i i I. Def. 22] NOTES ON DEFINITIONS 20—22 189 square as for a triangle (i.e. '* having one right angle "), this will not do in the case of the oblong, which, unless it were stated that (Aree of its angles are right angles, would not be sufficiently defined. If it be objected, as it was by Todhunter for example, that the definition of a square assumes more than is necessary, since it is sufficient that, being equilateral, it should have one right angle, the answer is that, as in other cases, the superfluity does not matter from Euclid's point of view ; on the contrary, the more of the essential attributes of a thing that could be included in its definition the better, provided that the existence of the thing defined and its possession of all those attributes is proved before the definition is actually used ; and Euclid does this in the case of the square by construction in i. 46, making no use of the definition before that proposition. The word rfunnbus (po/i^os) is apparently derived from ^cfijSoi^ to turn roufid and round^ and meant among other things a spinning-top. Archimedes uses the term solid rhombus to denote a solid figure made up of two right cones with a common circular base and vertices turned in opposite directions. We can of course easily imagine this solid generated by spinning \ and, if the cones were equal, the section through the common axis would be a plane rhombus, which would also be the apparent form of the spinning solid to the eye. The difficulty in the way of supposing the plane figure to have been named after the solid figure is that in Archimedes the cones forming the solid are not necessarily equal. It is however possible that the solid to which the name was originally given was made up of two equal cones, that the plane rhombus then received its name from that solid, and that Archimedes, in taking up the old name again, extended its signification (cf. J. H. T. Miiller, Beitrdge zur Terminologie der griechischen Mathematiker^ i860, p. 20). Proclus, while he speaks of a rhombus as being like a shaken, Le. deformed, square, and of a rhomboid as an oblong that has been moved, tries to explain the rhombus by reference to the appearance of a spinning square {r^pdyw^v jk}flpOVfL€VOv). It is true that the definition of a rhomboid says more than is necessary in describing it as having its opposite sides and angles equal to one another. The answer to the objection is the same as the answer to the similar objection to the definition of a square. Euclid makes no use in the Elements of the oblongs the rhombus^ the rhomboid^ and the trapezium. The explanation of his inclusion of definitions of the first three is no doubt that they were taken from earlier text-books. From the words ^'let quadrilaterals other than these be called trapezia," we may perhaps infer that this was a new name or a new application of an old name. As Euclid has not yet defined parallel lines and does not anywhere define a parallelogram^ he is not in a position to make the more elaborate classification of quadrilaterals attributed by Proclus to Posidonius and appearing also in Heron's Definitions. It may be shown by the following diagram, distinguishing seven species of quadnlaterals. Quadrilaterals parallelograms non-parallelograms rectangular non-rectangular two sides parallel no sides parallel I I (trapetium) (trapfioi4i) I — ' — I I • 1 I ' 1 square pbhn^ rhombus rhotubcid isosceles trapetium ualeuo trapeuum I I90 BOOK I [i. Deff. 33, S3 It will be observed that^ while Euclid in the above definition classes as trapezia all quadrilaterals other than iquares, obl(»igi, rfaombi, and rhomboid^ the word is in this classification restncted to quadrilaterals having two sides (only) parallel, and trapezoid is used to denote the rest £uclid appears to have used trapezium in the restricted sense of a quadrilateral with two sides parallel in his book xc^m ScoipccrcMr (on divisions of figures). Archimedes uses it in the same sense, but in one place describes it more precisely as a trapezium with its two sides parallel. Definition 23. napoAXiyAoi clcru^ tiSuax^ oTrtvcf If ny uifT^ hr a r A ff oSooi fcat Ik^oXAi^^mmu CIS air€tpov 1^* kicdrtpa ra yuifpii hn fiifiirMfio. uvf/ariwrowrw AAAijXaif. Parallel straight tines are straight Ufus whieh^ being in the same plane emd being produced indefinitely in bath Unctions^ do not meet one another in either dirution, IlapaAAi^Xof (alongside one another) written in one word does not appear in Plato ; but with Aristotle it was already a familiar term. €h aircifK>F cannot be translated ''to infinity" because these words might seem to suggest a r^n or place infinitely distant, whereas ck avcipor, which seems to be used indifferently with hr mipoy, is adverbial, meaning ''without limit," i.e. *' indefinitely." Thus the expr^sion is used of a magnitude beiog "infinitely divisible," or of a series of terms extending without limit In both directions^ c^* indrtfia, ra §i4pi^ literally *' towards both the parts" where *'parts" must be used in the sense of "regions" (cf Thuc ii. 96). It is clear that with Aristotle the general notion of parallels was that of straight lines which do not meet^ as in Euclid : thus Aristotle discusses the question whether to think that parallels do meet should be called a geometrical or an ungeometrical error {Anal, post, i. 12, 77 b 22), and (more interesting still in relation to Euclid) he observes that there is. nothing surprising in different hypotheses leading to the same error, as one might conclude that parallels meet by starting from the assumption, either (a) that the interior (angle) is greater than the exterior, or (b) that the angles of a triangle make up more than two right angles (Anal, prior. 11. 17, 66 a 11). Ajiother definition is attributed by Proclus to Posidonius, who said that ^^ parallel lines are tlwse which^ {being) in one plane^ neither converge nor diverge^ but have all the perpendiculars equal which are drawn from the points of one line to the other ^ while such (straight lines) as make the perpendiculars less and less continually do converge to one another ; for the perpendicular is enough to define (opt^ciF hdvaraC) the heights of areas and the distances between lines. For this reason, when the perpendiculars are equal, the distances between the straight lines are equals but when they become greater and less, the interval is lessened, and the straight lines converge to one another in the direction in which the less perpendiculars are" (Proclus, p. 176, 6—17). Posidonius' definition, with the explanation as to distances between straight { lines, their convergence and divergence, amounts to the definition quoted by ! Simplidus (an-Nairizi, p. 25, ed. Curtze) which described straight lines as | parallel if when they are produced indefinitely both ways^ the distance between themy or the perpendicular drawn from either of them to the other ^ is always equal and not different. To the objection that it should be prorved that the distance between two parallel lines is the perpendicular to them Simplicius •^1 1. Def. 23] NOTES ON DEFINITIONS 22, 23 191 replies that the definition will do equally well if all mention of the perpen- dicuiar be omitted and it be merely stated that the distance remains equal, although " for proving the matter in Question it is necessary to say that one straight line is perpendicular to both'' (an-Nairizi, ed. Besthom-Heiberg, p. 9). He then quotes the definition of "the philosopher Aganis": ^^Parailei straight Ones are straight lines^ situated in the same plane^ the distance between whichj if they are produced iiidefiniidy in both directions at the same time^ is everywhere the sameJ' (This definition forms the basis of the attempt of "Aganis" to prove the Postulate of Parallels.) On the definition Simplicius remarks that the words ''situated in the same plane'' are perhaps unnecessary, •since, if the distance between the lines is eve^where the same, and one does not incline at all towards the other, they must for that reason be in the same plane. He adds that the ''distance" referred to in the definition is the shortest line which joins things disjoined.. Thus, between point and point, the distance is the straight line joining them ; between a point and a straight line or between a point and a plane it is the perpendicular drawn from the pomt to the line or plane; "as regards the distance between two lines, that distance b, if the lines are parallel, one and the same, equal to itself at all places on the lines, it is the shortest distance and, at all places on the lines, perpendicular to both " (i^k/. p. 10). The same idea occurs in a quotation by Proclus (p. 177, 11) from Geminus. As part of a classification of lines which do not meet he observes : " Of lines which do not meet, some are in one plane with one another, others not. Of those which meet and are in one plane, sofne are always the same distance from one another^ others lessen the distance continually, as the hyper- bola (approaches) the straight line, and the conchoid the straight line (i.e. the asymptote in each case). For these, while the distance is being continually lessened, are continually (in the position oQ not meeting, though they converge to one another ; they never converge entirely, and this is the most paradoxical theorem in geometry, since it shows that the convergence of some lines is non- convergent. But of lines which are always an equal distance apart, those which are straight and never make the (distance) between them smaller, and which are in one plane, are parallel." Thus the eguidistance-thieory of parallels (to which we shall return) is very fully represented in antiquity. I seem also to see traces in Greek writers of a conception equivalent to the vicious directian-Htieory which has been adopted in so many modem text-books. Aristotle has an interesting, though obscure, allusion in Ana/, pricr. 11. 16, 65 a 4 to a, petitio principii committed by "those who think that diey draw parallels " (or " establish the theory of parallels," which is a possible translation of ra« ira^KiXAi^Xov« ypa^civ): "for they un- consciously assume such things as it is not possible to demonstrate if parallels do not exist" It is clear from this that there was a vicious circle in the then current theory of parallels ; something which depended for its truth on the properties of parallels was assumed in the actual proof of those properties, e.g. that the three angles of a triangle make up two right angles. This is not the case in Euclid, and the passage makes it clear that it was Euclid himself who got rid of the petitio principii in earlier text-books by formulating and premising before i. 39 the famous Postulate 5^ which must ever be r^arded as among the most epoch-making achievements in the domain of geometry. But one of the commentators on Aristotle, Philoponus, has a note on the above passage purporting to give the specific character of the petitio principii alluded to; and it is here that a direction-theory of parallels may be hinted at. 192 BOOK I [i. DBr. 13 whether Philoponus is or is not rig^t in supposing Aat this was what Aristotle had in mind. Philoponus says: ''The same thinff is done by those iriio dnw parallels, namely begging the original question; for thc^ will have it tfiat it is possible to draw parallel straight lines from the mendian circle, and they assume a point, so to say, falluiff on the plane of that circle and thus they draw the straight lines. And what was sought is thereby assumed; for he who does not admit the genesis of the parallels will not admit the point referred to either." What is meant is, I think, somewhat as follows. Given a straight line and a point through which a parallel to it is to be drawn, we are to suppose the given straight line placed in the plane of the meridian. J Then we are told to draw through the given point anodier straight line in the ^ plane of the meridian (strictly speaking it should be drawn in a plane pualld k to the plane of the meridian, but the idea is that, compared with the sixe of | the meridian circle, the distance between the point and the straight line is { negligible) ; and this, as I read Philoponus, is supposed to be equivalent to j assuming a very distant point in the meridian plane and joining the given J point to it But obviously no ruler would stretch to such a pomt, and the objector would say that we cannot really direct a straight line to die assumed distant point except by drawing it, widiout more ado, paralld to the given straight line. And herein is the pttiHo frmcifii, I am confirmed in seeing in Philoponus an allusion to a dineiiOH'thearj by a remark of Schotten on a similar reference to the meridian plane supposed to be used by advocates of . that theory. Schotten is aiguing that direction is not in itself a conception ' such that you can predicate Me direction of two different lines. ** If any one should reply that nevertheless many lines can be conceived which aU have the direction from north to south^^ he replies that this represents only a nominal, not a real, identity of direction. Coming now to modem times, we may classify under three groups practically all the different definitions that have been given of parallels (Schotten, op, cit. 11. p. 188 sqq.). (i) Parallel straight lines hm*e no point common^ under which general conception the following varieties of statement may be included : {a) they do not cut one another^ ^ {b) they meet at infinity^ or ^ (c) they have a common point at infinity. \ (2) Parallel straight lines have the same^ or lihe^ direction or directions^ under which class of definitions must be included all those which introduce ' transversals and say that the parallels mahe equal angles with a transverscU. . (3) Parallel straight lines have the distance t^etween thetn constant \ \ with which group we may connect the attempt to explain a parallel as the > geometrical locus of all points which are equidistant from a straigkt line. { But the three points of view have a good deal in common ; some of them lead easily to the others. Thus the idea of the lines having no point common led to the notion of their having a common point at infinity, through the | influence of modem geometry seeking to embrace different cases under one f conception ; and then again the idea of the lines having a common point at ^ infinity might suggest their having the same direction. The " non-secant " f idea would also naturally lead to that of equidistance (3), since our observation shows that it is things which come nearer to one another that tend to meet, and hence, if lines are not to meet, the obvious thing is to see that they shall not come nearer, i.e. shall remain the same distance apart. \ I I. D«F. 23] NOTE ON DEFINITION 23 I93 We will now take the three groups in order. (i) The first observation of Schotten is that the varieties of this group which regard parallels as (a) meeting at infinity or (d) having a common point at mfinity (first mentioned apparentlv by Kepler, 1604, as a '*fa^on de parler" and then used by Desaigues, 1639) are at least unsuitable definitions tor elementary text-books. How do we know that the lines cut or meet at infinity? We are not entitled to assume either that they do or that they do not, because "infinity'' is outside our field of observation and we cannot verify either. As Gauss says (letter to Schumacher), " Finite man cannot claim to be able to regard the infinite as something to be grasped by means of ordinary methods of observation." Steiner, in speaking of the rays passing through a point and successive points of a straight line, observes that as the point of intersection gets further away the ray moves continually in one and the same direction ("nach einer und derselben Richtung hin"); only in one position, that in which it is parallel to the straight line, "there is no real cutting^* between the ray and the straight line ; what we have to say is that the ray is *^ directed towards the infinitely distant paint an the straight line.^^ It is true that higher geometry has to assume that the lines do meet at infinity: whether such lines exist in nature or not does not matter (just as we deal with "straight lines " although there is no such thing as a straight Une). But if two lines do not cut at any finite distance, may not the same thing be true at infinity also ? Are lines conceivable which would not cut even at infinity but always remain at the same distance from one another even there ? Take the case of a line of railway. Must the two rails meet at infinity so that a train could not stand on them there (whether we could see it or not makes no difference)? It seems best therefore to leave to higher geometry the conception of infinitely distant points on a line and of two straight lines meeting at infinity, like imaginary points of intersection, and, for the purposes of elementaiy geometry, to rely on the plain distinction between "parallel" and "cutting" which average human mtelligence can readily grasp. This is the method adopted by Euclid in his definition, which of course belongs to the group (i) of definitions regarding parallels as non-secant It is significant, I think, that such authorities as Ingrami {Elementi di geometria^ 1904) and Enriques and Amaldi (Elementi di geometria^ 1905)1 after all the discussion of principles that has taken place of late years, give definitions of parallels equivalent to Euclid's : " those straight lines in a plane which have not any point in common are called parallels." Hilbert adopts the same point of view. Veronese, it is true, takes a different line. In his great work Fondamenti di geametria, 1 891, he had taken a ray to be parallel to another when a point at infinity on the second is situated on the first ; but he appears to have come to the conclusion that this definition was unsuitable for his Elementi. He avoids however giving the Euclidean definition of parallels as "straight lines in a plane which, though produced indefinitely, never meet," because "no one has ever seen two straight lines of this sort," and because the postulate generally used in connexion vnth this definition is not evident in the way that, m the field of our experience, it is evident that only one straight line can pass through two points. Hence he gives a different definition, for which he claims the advantage that it is independent of the plane. It is based on a definition of figures " opposite to one another with respect to a point" (or reflex figures). "Two figures are opposite to one another with respect to a point C7, e.g. the figures ABC ... and A'ffC.,.^ M to every point of the one there corresponds one sole point of the other, and if the segments H. s. 13 t94 6001C t [i. DBF. tj OA^ OB^ OCy ... joining the points of one figure to C7 are respectively equal and opposite to the segments 0A\ OB^ 0C\ ... joining to O the corresponding points of the second " : then, a iramsverstU of two straight lines being any segment having as its extremities one point of one line and one point of the other, " two straight lines are aUlei parallel if one of them contains two foinis opposite to two points of the other with reject to the middle point of a common transverscU,^^ It is true, as Veronese says, that the paiallek so defined and the parallels of Euclid are in substance the same; but it can hardly be said diat the definition gives as good an idea of the essential nature of parallels as does Euclid's. Veronese has Xjoprove^ of course, that his parallels have no point in common, and his ''Postulate of Parallds" can hardly be called more evident than Euclid's : " If two straight lines are parallel, they are figures opposite to one another with respect to the middle points of all their transversal segments." (2) The direction-xheoiy. The fallacy of this theory has nowhere been more completely exposed than by C. L. Dodgson (EuJid and his modem Moals, 1879). According to Killing {Einfuhrung in die Grundlagm der Geometrie^ i. p. 5) it would appear to have originated with no less a person than Leibniz. In the text-books which employ this method the notion of direetion appears to be regarckd as a primary, not a derivative notion, since no definition is given. But we ought at least to know how the same direction or like directions can be recognised when two different straight lines are in question. But no answer to this question is forthcoming. The fact is that the whole idea as applied to non- coincident straight lines is derived firom knowledge of the properties of parallels ; it is a case of explaining a thing by itself. The idea of parallels being in the same direction perhaps arose from the conception of an angle as a differefue of direction (the hoUowness of which has already been expmed) ; sameness of direction for parallels follows from the same "difference of direction" which both exhibit relatively to a third line. But this is not enough. As Gauss said {Werhe, iv. p. 365), " If it [identity of direction] is recognised by the equality of the angles formed with one third straight bne, we do not yet know vnthout an antecedent proof whether this same equality will also be found in the angles formed with a fourth straight line " (and any number of other transversals) ; and in order to make this theory of parallels valid, so far from getting rid of axioms such as Euclid's, you would have to assume as an axiom what is much less axiomatic, namely that "straight lines which make equal corresponding angles vnth a certain transversal do so with any transversal" (Dodgson, p. loi). (3) In modem times the conception of parallels as equidistant straight lines was practically adopted by Clavius (the editor of Euclid, bom at Bamberg, 1537) and (according to Saccheri) by Borelli {Euclides restitutus^ 1658) although they do not seem to have defined parallels in this way. Saccheri points out that, before such a definition can be used, it has to be proved that " the geometrical locus of points equidistant fi^m a straight J line is a straight line." To do him justice, Clavius saw this and tried to prove it: he makes out that the locus is a straight line according to the definition of Euclid, because "it lies evenly with respect to all tiie points on it"; but there is a confusion here, because such "evenness" as the locus has is with respect to the straight line fi^m which its points are equidistant, and there is noUiing to show that it possesses this property with respect to itself. In fact the theorem cannot be proved vrithout a postulate. J I I. Post, i] N0T£ ON t>OStULATfi i tpj Postulate i. HinioBia dwo toitos arjfjLtlov hci xav (rqfUiov cMciav ypafjLfirjy ayayctv. Let the following be postulated: to draw a straight line from any point to any point. From any point to any point. In general statements of this kind the Greeks did not say, as we do, "a/iy point," ^*any triangle" etc., but ^* every point," ^^ every triangle" and the like. Thus the words are here literally "from every point to every point." Similarly the first words of Postulate 3 are " with every centre and distance," and the enunciation, e.g., of I. 1 8 is " In every triangle the greater side subtends the greater angle." It will be remembered that, according to Aristotle, the geometer must in general assume what a thing is, or its definition, but must prove that it is, i.e. the existence of the thing corresponding to the definition : only in the case of the two most primary things, points and lines, does he assume, without proof, both the definition and the existence of the thing defined. Euclid has mdeed no separate assumption affirming the existence oi points such as we find nowadays in text-books like those of Veronese, Ingrami, Enriques, "there exist distinct points" or "there exist an infinite number of points." But, as re- gards the only lines dealt with in the Elements^ straight lines and circles, existence is asserted in Postulates i and 3 respectively. Postulate i however does much more than (i) postulate the existence of straight lines. It is (2) an answer to a possible objector who should say that you cannot, with the imperfect instruments at your disposal, draw a mathematical straight line at all, and consequently (in the words of Aristotle, Anal, post. i. 10, 76 b 41) that the geometer uses ^se hypotheses, since he calls a line a foot long when it is not or straight when it is not straight. It would seem (if Gherard's translation is right) that an-Nairizi saw that one purpose of the Postulate was to refute this criticism : " the utility of the first three postulates is (to ensure) that the weakness of our equipment shall not prevent (scientific) demonstration" (ed. Curtze, p. 30). The fact is, as Aristotle says, that the geometer's demon- stration is not concerned with the particular imperfect straight line which he has drawn, but with the ideal straight line of which it is the imperfect represer^tation. Simplicius too indicates that the object of the Postulate is rather to enable the drawing of a mathematical straight line to be imagined than to assert that it can actually be realised in practice : " he would be a rash person who, taking things as they actually are, should postulate the drawing of a straight line from Aries to Libra." There is still something more that must be inferred from the Postulate combined with the definition of a straight line, namely (3) that the straight line joining two points is unique : in other words that, tf tivo straight lines ("rectilineal segments," as Veronese would call them) luwe the same extremities^ - they must coinade throughout their length. The omission of Euclid to state this in so many words, though he assumes it in i. 4, is no doubt answerable for the interpolation in the text of the equivalent assumption that two straight lines cannot enclose a space^ which has constantly appeared in mss. and editions of Euclid, either among Axioms or Postulates. That Postulate i included it, by conscious implication, is even clear from Proclus* words in his note on l 4 (p. 239, 16) : "therefore two straight lines do not enclose a space, and it was with knowledge of this fact that the writer of the Elements said in the first of his Postulates, to draw a straight line from any point to any pointy implying that it is ^M^ straight line which would always join the two points, not two.^ 13— » I 19^ BOOK I [l Post. i»s Proclus attempts in the same note (p. 339) to P^yoe that two straight lines cannot enclose a space, using as his bws the definition of the diameter of a circle and the theorem, stat^ in it, that any diameter divides tfie cirde into two equal parts. ' Suppose, he says, ACB^ ADB to be two straight lines enclosing a q»oe. Produce them (beyond B) indefinitely. With centre B and distance AB describe a circle, cutting the lines so produced va F^ E respectively. Then, since ACBF, ADBE are both diameters cutting off semi-circles, the arcs AE^ AEF are equal : which is impossible. Therefore etc. It ¥nll be observed, however, that the straight lines produced are assumed to meet the circle given in two different points E^ F^ whereas, for anything we know, Ey F might coincide and the straight lines have three common points. The proof is therefore delusive. Saccheri gives a different {xoof. From Euclid's definition of a straight line as that which lies evenly with its points he infers that, when such a line is turned about its two extremities, which remain fixed, all the points on it must remain throughout in the same position, and cannot take up different positions as the revolution proceeds. " In this view of the straight line the truth of the assertion that two straight lines do not enclose a space is obviously involved. In fact, if two lines are given which enclose a s^ce, and of which the two points A and X are the common extremities, it is easily shown that neither, or else only one, of the two lines is straight" It is however better to assume as a postuiate the fact, inseparably connected with the idea of a straight line, that there exists onfy one straight line containing two given points^ or, if two straight lines have two points in commotiy they coiticide throughout. Postulate 2. Kal icaetpaxT\kivyfv tMtiav Kara to (twc^^ i'f* cMctac ^ic/SaXciy. To produce a finite straight line continuously in a straight line, I translate icvtrmaxr\tivrjpf by finite^ because that is the received equivalent, and because any alternative word such as limited^ terminated^ if applied to a straight line, would equally fail to express what modem Italian geon^eters aptly call a rectilineal segment^ that is, a straip;ht line having two extremities. Just as Post. I asserting the possibility of drawing a straight line from any one point to another must be held to declare at the same time that the straight line so drawn is unique, so Post 2 maintaining the possibility of producing a finite straight line (a "rectilineal segment ") continuously m a straight hne must also be held to assert that the straight line can only be produced in one way at either end, or that the produced part in either direction is unique ; in other words, that two straight lines cannot have a common segment. This latter assumption is not expressly appealed to by Euclid until xi. i. But it is needed at the very beginning of Book'i. Proclus (p. 314, 18) says that 2^no of Sidon, an Epicurean, maintained that the very first proposition i. i requires it to be admitted that *' two straight lines cannot have the same segments " ; otherwise AQ BC might meet before they arrive at C and have the rest of their length common, in which case the actual triangle formed by them and AB would not be equilateral. The assumption that two straight lines cannot have a common segment is certainly necessary in I. 4, where one side of one triangle is placed on that side of the other ( j L Post. 2] NOTES ON POSTULATES i. 2 197 triangle which is equal to it, and it is inferred that the two coincide throughout their length : this would by no means follow if two straight lines could have a common segment Proclus (p. 215, 24), while observing that Post. 2 clearly indicates that the produced portion must be one^ attempts to prove it, but unsuccessfully. Both he and Simplicius practically use the same argument. Suppose, says Proclus, that the straight lines ACy AD have AB as a common segment. With centre B and radius BA describe a circle (Post 3) meeting AC^ AD in C, D. Then, since ABC is a straight line through the centre, AEC is a semi-circle. Similarly, ABD being a straight line through the centre, AED is a semi-circle. Therefore AEC is equal to AED\ which is impossible. Proclus observes that 2^no would object to this proof as really depending on the assumption that "two circumferences (of circles) cannot have one portion common " ; for this, he would say, is assumed in the common proof by superposition of the fact that a circle is bisected by a diameter, since that proof takes it for granted that, if one part of the circumference cut off by the diameter, when applied to the other, does not coincide with it, it must neces- sarily fall either entirely outside or entirely inside it, whereas there is nothing to prevent their coinciding, not altogether, but in part only ; and, until you really prove the bisection of a circle by its diameter, the above proof is not valid. ' Posidonius is represented as having derided Zeno for not seeing that the proof of the bisection of a circle by its diameter goes on just as well if the circumferences fail to coincide in part only. But the true objection to the proof above given is that the proof of the bisection of a circle by any diameter i/f^'^ assumes that two straight lines cannot have a common segment; for, if we wish to draw the diameter of a circle which has its extremity at a given point of the circumference we have to join the latter point to the centre (Post i) and then to produce the straight line so drawn till it meets the circle again (Post 2), and it is necessary for the proof that the produced part shall be unique. Saccheri adopted the proper order when he gave, first the proposition that two straight lines cannot have a common s^ment, and after that the proposition that any diameter of a circle bisects the circle and its circumference. Saccheri's proof of the former is very interesting as showing; the thorough- ness of his method, if not at the end entirely convincing. It is in five stages which I shall indicate shortly, giving the full argument of the first only. Suppose, if possible, that AX is a common segment of both the straight lines AXB^ AXC^ in one plane, produced beyond X. Then describe about X as centre, with radius XB or XC^ the arc BMC^ and draw through X to any point on it the straight line XM. (1) I maintain that, with the assumption made, the line AXM is also a straight line which is drawn from the point A to the point X and pro- duced beyond X. For, if this line were not straight, we could draw another straight line AM which for its part would be straight This straight line will either (a) cut one of the two straight lines XB^ XC in a certain point K or (b) enclose one of them, for instance XBy in the area bounded by AX^ XMsind APLM. 193 BOOK I [lPost. i But the first alternative (a) obviously contndicU the forq^oiiig lemma rtliat two straight lines cannot enclose a space^ since in that case the two lines AXJ^, ATK^ which by hypothesis are straight, would enclose a space. The second possibility ijf) is at once seen to invdve a similar abaurdity. For the straight line XB must, when produced beyond B^ ultimatel]r meet APLM in a point Z. Consequently the two lines AXBL^ APL^ which by hypothesis are straight, would again enclose a space. If however we were to assume that the straight line XB produced beyond B will ultimately meet either the straight line XMor the straight line XA in another point, we should in the same way arrive at a contradiction. From this it obviously follows that, on the assumption made, the line AXM\% itself the straight line which was drawn from the point ^ to the point M'y and that is what was maintained. The remaining stages are in substance these. (ii) Iftke straight line AXB, regarded as rigid^ revohes about AX as axis^ U cannot assume two more positions in the same fiane^ so tkat^ for exam^e^ in one position XB should coincide with XC, and in the other with XM. ^ [This is proved by considerations of symmetry. AXB cannot be altogether ** similar or equal to " AXC^ if viewed from the same side (left or ri^t) of both : otherwise they would coincide, which by hypothesis they do not But there is nothing to prevent AXB viewed from one side (sav the left) being '* similar or equal to '' AXC viewed from the other side (i.e. the right), so that AXB can^ without any chanffe, be brought into the position AXC. AXB cannot however tale the position of the other straight line AXAitm i well If they were like on one side, they would coincide; if they were like on 1 opposite sides, AXM^ AXC would be like on the same side and therefore j comdde.] I (iii) The other positions of AXB during the revolution must be above or 4 below the original plane. 4 (iv) It is next maintained that there is a point l^ on the arc BC such that^ if 1 XD is drffwn^ AXD is not only a straight line but is such that viewed from the left I side it is exactly ^^ similar or equaV^ to what it is when viavedfrom the right side, > {Firsts it is proved that points M^ Fcan be found on the arc, corresponding \ in the same way as B^ C do, but nearer together, and of course AXM^ AXF are both straight lines. Secondly^ similar corresponding points can be found still nearer together, and so on continually, until either (a) we come to one point D such that AXD \ is exactly iihe itself when the right and left sides are compared^ or (b) there are ; two ultimate points of this sort M^ F^ so that both AXM^ AXF have this property. Thirdly^ (b) is ruled out by reference to the definition of a straight line. ' Hence (a) only is true, and there is only one point D such as described.] (v) Lastly, Saccheri concludes that the straight line AXD so determined i *' is alone a straight line, and the immediate prolongation from A beyond X to />," relying again on the definition of a straight line as 'Mying evenly." Simson deduced the proposition that two straight lines cannot have a common segment as a corollary from i. 1 1 ; but his argument is a complete petitio principii^ as shown by Todhunter in his note on lliat proposition. Produs (p. SI 7, lo) records an ancient proof also based on the proposition I. 1 1. Zeno, he says, propounded this proof and then critidsed it I. Post- 2, 3] NOTES ON POSTULATES 2, 3 199 Suppose that two straight lines AC^ AD have a common segment AB^ and let BE be drawn at right angles to AC. Then the angle BBC is right. If then the angle EBD is also right, the two angles will be equal : which is impossible. If the angle EBD is not right, draw ^^at right angles to AD \ therefore the angle FBA is right. But the angle EBA is right Therefore the angles EBA^ FBA are equal : which is impossible. Zeno objected to this, says Proclus, because it assumed the later pro- position I. 1 1 for its proof. Posidonius said that there was no trace of such a proof to be found in the text-books of Elements, and that it was only invented by Zeno for the purpose of slandering contemporary geometers. Posidonius maintains further that even this proof has something to be said for it. There must be some straight line at right angles to each of the two straight lines A C, AD (the very definition of right angles assumes this): ^^ suppose /Aen U happens to be the straight line we have set upJ* Here then we have an ancient instance of a defence of hypothetical construction^ but in such apologetic terms (" it is possible to say something even for this proof") that we may conclude that in general it would not have been accepted by geometers of that time as a Intimate means of proving a proposition. Todhunter proposed to deduce that two straight lines cannot have a common segment from i. 13. But this will not serve either, since, as before mentioned, the assumption is really required for i. 4. It is best to make it a postulate. Postulate 3. Kal Tovrl iccFrpcp jcal Zuurrrnijari kvkXjO¥ ypa^cotfai. To describe a circle with any centre and distance. In this case Euclid's Jext has the passive of the verb: '*a circle can be drawn " ; Proclus however has the active (ypJ^oi) as Euclid has in the first two Postulates. Distance^ iuKm/fiaTL This word, meaning ** distance *' quite generally (cf. Arist Metaph. 1055 a 9 *'it is between extremities that distance is greatest," ibid. 1056 a 36 '' things which have something between them, that is, a certain distance "), and also " distance " in the sense of '' dimension " (as in " space has three dimensions, length, breadth and depth," Arist. Physics iv. i, 209 a 4), was the regular word used for describing a circle with a certain radius^ the idea being that each point of the circumference was at that distance from the centre (cf. Arist. Meteorologica in. 5, 376 b 8 : "if a circle be drawn. ..with distance Mil "). The Greeks had no word corresponding to radius : if they had to express it, they said "(straight lines) drawn from the centre" (al Ik rov Kivrpov, Eucl. III. Def. I and Prop. 26; Meteorologica 11. 5, 362 b i has the full phrase ol Ik tov mtyrpov dyofuyiu ypofifjuoi), Mr Frankland observes that it would be remarkable if, unlike Postulates i and 2, this Postulate implied merely what it says, that a circle can be drawn with any centre and distance. We may r^ard it, if we please, as helping to the complete delineation of the Space which Euclid's geometry is to investigate formally. The Postulate has the effect of removing any restriction upon the size of the circle. It may (i) be indefinitely small, and this implies that space is continuous, not discrete, with an irreducible minimum distance between soo BOOK I [l Post 3, 4 contiguous pomts in it (a) The drde may be indefinitely bu8e» wfaidi implies the fundamental hypothecs of h^miudi of space. This last assumed characteristic of space is essential to the proof of i. i6^ a theorem not universally valid in a space whidi is unbounded in extent but finite in sise. It would however be unsafe to suppose that Euclid foresaw the use to which his Postulate might thus be put, or formulated it with such an intention. Postulate 4. Koi irauiif ths 6p$a9 yta^tom Itrat oXXipXaif Aoi. ITiat all right angles are equal to 4me another. While this Postulate asserts the essential truth that a risht angle is a determinate magnitude so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen firom the following consideratioiL ^ If the statement is to be proved^ it can only be proved b^ the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the invaruMlity ofjigiirest which would therefore have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal; and hence his postulate must be tidten as equivalent to me principle of invariability of figures or its equivalent, the homogeneity of space. According to Proclus, Geminus held that this Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some amstntction but expresses an essential property of right angles. Produs further observes (p. 188, 8) that it is not a postulate in Aristotle's sense either. (In this I think he is wrong, as exjdained above.) Proclus himself, while regarding the assumption as axiomatic (** the equality of right angles suggests itself even by virtue of our common notions"), is prepared with a proof, if such is asked for. Let ABC, DEF be two right angles. If they are not equal, one of them must be the greater, say ABC. Then, if we apply DE to AB, EF H- will fall within ABC, as BG. Produce CB to H. Then, since ABC is a right angle, so is ABH, and the two angles are equal (a right angle being by definition equal to its adjacent angle). Therefore the angle ABH\% greater tlum the angle ABG. Producing GB to K, we have similarly the two angles ABK, ABG both right and equal to one another; whence the angle ABHS& less than the angle ABG. But it is also greater : which is impossible. Therefore etc. A defect in this proof is the assumption that CB, GB can each be produced only in one way, and that ^9^ falls outside the angle ABH. Saccheri's proof is more careful in that he premises a third lemma in addition to those asserting (i) that two straight lines cannot enclose a space and (2) that two straight lines cannot have a common segment The third lemma is : If two straight Unes AB, CXD meet one another at an intermediate point X, they do not touch at that point, but cut one another. 1 i I 1. Post. 4] NOTES ON POSTULATES 3, 4 aoi M Suppose now that DA standing on BAC makes the two angles DAB^ Z>^C equal, so that each is a right angle by the definition. Similarly, let LHioxm with the straight line FHM the right angles LHFy LHM. Let DA^' HL be equal ; and sup- pose the whole of the second figure so laid upon the first that the point ^falls on A^ and L on Z>. Then the straight line FHMmW (by the third lemma) not touch the straight line BC at A ; it will either (a) coincide exactly with BC, or (d) cut it so that one of its extremities, as F, will fall above [BC] and the other, M^ below it. If the alternative (a) is true, we have already proved the exact equality of all rectilineal right angles. Under alternative (b) we prove that the angle LHF, being equal to the angle DAF, is less than the angle DAB or DAC, and a fortiori less than the angle DAM ox LHM\ which is contrary to the hypothesis. [Hence (a) is the only possible alternative, so that all right angles are equal.] Saccheri adds that it makes no difference if the angle DAF diverges infinitely little from the angle DAB, This would equally lead to a conclusion contradicting the hypothesis. It will be observed that Saccheri speaks of ''the exact equality of all rectilineal right angles." He may have had in mind the remark of Pappus, quoted by Proclus (p. 189, 11), that the converse of this postulate, namely that an angle which is equal to a right angle is also right, is not necessarily true, unless the former angle is rectilinecU. Suppose two equal straight lines BA^ BC9X right angles to one another, and semi-circles described on BA, BC respectively as AEB, BDC in the figure. Then, since the semi-circles are equal, tiiey coincide if applied to one another. Hence the ''angles ** EBA, DBC are equal Add to each the " angle " ABD ; and it follows that the lunular angle EBD is equal to the right angle ABC. (Similarly, if BA, BC be inclined at an acute or obtuse angle, instead of at a rfght angle, we find a lunular angle equal to an acute or obtuse angle.) This is one of the curiosities which Greek commentators delighted in. Veronese, Ingrami, and Enriques and Amaldi deduce the fact that all right angles are equal from the equivalent fact that all fiat angles are equals which is either itself assumed as a postulate or immediately deduced from some other postulate. HUbert takes quite a different line. He considers that Euclid did wrong in placing Post 4 among "axioms." He himself, after his Group in. of Axioms containing six relating to congruence, proves several theorems about the congruence of triangles and angles, and then deduces our Postulate. As to the raison iPitre and the place of Post 4 one thing is quite certain. It was essential from Euclid's point of view that it should come before Post. 5, since the condition in the latter that a certain pair of angles are together less than two right angles would be useless unless it were first made clear Ihat right angles are angles of determinate and invariable magnitude. ao3 BOOK I [i. Post. 5 1 Postulate 5. Kcu iay etc Svo c^ctat cMcia i§iwiwr€vaa rkt hrri^ mu hn rk aira fU/ni vtMit Suo 6ft6mv iXaaaoya^ iroc^ iKPaXXofiiva% r&« Svo cMctat Jv* avcipor ov/cviVTCcr, 7>(a/, (/*« straight line falling on two straighl lines make ike interior angles on the same side less than two right angles^ the two straight Knes^ if prodneed indefinitely^ meet on that side on which an the angles less than the two rfght angles. Although Aristotle gives a dear idea of what he understood by a postulaie^ he does not give any instances from g^metry; still less has he an^ allusion recalling the particular postulates found in Euclid. We naturally infer that the formulation of these postulates was Euclid's own work. There is a more positive indication of the originality of Postulate 5* since in the passage {Anal, prior. II. 16, 65 a 4) quoted above in the note on the definition of parallels he alludes to some petitio primipii involved in the theory of parallels current in his time. This reproach was removed by Euclid when he laid down this epoch-making Postulate. When we ocmsider the countless successive attempt made through more than twenty centuries to prove the Postulate, many of them by geometers of abili^, we cannot but admire the genius of the man who conduded that such a hypothesisi which 'he found necessary to the validity of his whole system of geometiy, wasjc^ly indem(mstmble. From the very b^mning» as we know irom ProcIus| the Tostulate was attacked as such, and attempts were made to prove it as a theorem or to get rid of it by adopting some odier definition of jparallds; while in modem times the literature of the subject is enormous. Riccardi {^aggju^ di una bibU^^fta Euclidea^ Part iv., Bologna, 1890) has twenty quarto pages of titles of mono- graphs relating to Post 5 between the dates 1607 and 1887. Max Simon \Ueber die Entwichlung der Elementar-geometrie im XIX. Jahrhundtrt^ 1906) notes that he has seen three new attempts, as late as 1891 (a century after Gauss laid the foundation of non-Euclidean geometry), to prove the theory of ' parallels independently of the Postulate. Max Simon himself (pp. 53 — 61) gives a large number of references to books or articles on the subject and refers to the copious information, as to contents as wdl as names, con- tained in Schotten's Inhalt und Methodt des planimetrischen Unterrichts^ IL PP- 1*3— 33»- This note will include some account of or allusion to a few of the most noteworthy attempts to prove the Postulate. Only those of andent times, as being less generally accessible, will be described at any length; shorter references must suffice in the case of the modem geometers who have made the most important contributions to the discussion of the Postulate and have thereby, in particular, contributed most towards the foundation of the non- Euclidean geometries, and here I shall make use prindpally of the valuable Article 6, Sulla teoria delle parallele e sulk geomeirie nem-euclidee (by Roberto Bonola), in Ouestioni riguardanti la geametria elementare (pp. 143 — 222). Proclus (p. 191, 21 sqq.) states very clearly the nature of the first objections taken to the Postulate. *'This ought even to be struck out of the Postulates altogether; for it is a theorem involving many difficulties, which Ptolemy, in a certain book, set himsdf to solve, and it requires for the demonstration of it a number of definitions as well as the(»ems. And the converse of it is actually pioved by Eudid himsdf as a theorem. It may be that some would be I I. Post. 5] NOTE ON POSTULATE 5 203 1 deceived and would think it proper to place even the assumption in question among the postulates as affording, in the lessening of the two right angles, ground for an instantaneous belief that the straight lines converge and meet. To such as these Geminus correctly replied that we have learned from the very pioneers of this science not to have any regard to mere plausible imagin- ings when it is a question of the reasonings to be included in our geometrical doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of a rhetorician as to accept mere plausibilities from a geometer; and Simmias is made by Plato to say that he recognises as quacks those who fashion for themselves proofs from probabilities. So in this case the fact that, when the right angles are lessened, the straight lines converge is true and necessary; but the statement that, since they converge more and more as they are pro- duced, they will sometime meet is plausible but not necessary, in the absence of some argument showing that this is true in the case of straight lines. For the fact that some lines exist which approach indefinitely, but yet remain non-secant (dav/LnrrcDroi), although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines. May not then the same thing be possible in the case of straight lines which happens in the case of the lines referred to ? Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned (hypothesis) ? "It is then clear from this that we must seek a proof of the present theorem, and that it is alien to the special character of postulates. But how it should be proved, and by what sort of arguments the objections taken to it should be removed, we must explain at the point where the writer of the Elements is actually about to recall it and use it as obvious. It vnll be necessary at that stage to show that its obvious character does not appear independendy of proof, -but is turned by proof into matter of knowledge." Before passing to the attempts of Ptolemy and Proclus to prove the Postulate, I should note here that Simplicius says (in an-NairIzi, ed. Besthom- Heiberg, p. 119, ed. Curtze, p. 65) that this Postulate is by no means manifest, but requires proof, and accordingly '* Abthiniathus " and Diodorus had already proved it by means of many different propositions, while Ptolemy also had explained and proved it, using for the piffpose Eucl. i. 13, 15 and 16 (or 18). The Diodorus here mentioned may be the author of the Analemma on which Pappus wrote a commentary. It is difficult even to frame a conjecture as to who " Abthiniathus " is. In one place in the Arabic text the name appears to be written " Anthisathus " (H. Suter in Zeitschrift fiir Math, und Pkysiky xxxviii.y hist, litt Abth. p. 194). It has occurred to me whether he might be Peithon, a friend of Serenus of Antinoeia (Antinoupolis) who was long known as Serenus of Antissa, Serenus says (De sectione cylindri^ ed. Heiberg, p. 96): "Peithon the geometer, explaining parallels in a work of his, was not satisfied with what Euclid said, but showed their nature more cleverly by an example; for he says that parallel straight lines are such a thing as we see on walls or on the ground in the shadows of pillars which are made when either a torch or a lamp is burning behind them. And, althougn this has only been matter of merriment to every one, I at least must not deride it, for the respect I have for the author, who is my friend.'' If Peithon was known as " of Antinoeia " or " of Antissa,** the two forms of the mysterious name might perhaps be an attempt at an equivalent; but this is no more than a guess. 304 BOOK I [i. Post. 5 Simplicius adds in full and word for word the attempt of his "friend** or his *' master Aganis " to prove the Postulate. Proclus returns to the subject (p. 365, 5) in his note on EucL i. 39. He says that before his time a certain numb^ of g^meters had classed as a theorem this Euclidean postulate and thought it matter for proof, and he then proceeds to give an account of Ptolem/s argument Noteworthy attempts to prove the Postulate. Ptolemy. We learn from Proclus (p. 365, 7 — 11) that Ptolemy wrote a book on the proposition that " straight lines drawn from angles less than two right angles meet if produced," and that he used in his ''proof" many of the theorems in Euclid preceding i. 29. Proclus excuses himself from reproducing the nrly part of Ptolemy's argument, only mentioning as one of the propositions proved in it the theorem of EucL i. a8 that, if two straight lines meeting a transversal make the two interior angles on the same side equal to two ri^t angles, the straight lines do not meet, however far produced. I. From Proclus' note on 1. 28 (p. 362, 14 sq.) we know that Ptdemy proved this somewhat as follows. Suppose that there are two straight lines AB, CD, and that EFGH, meeting them, makes the angles BFG, FGD equal to two right angles. I say that AB, CD are paralH that is, they are non-secant For, if possible, let FB, GD meet at K. Now, since the angles BFG, FGD are equal to two right angles, while the four angles AFG, BFG, FGD, FGC are together equal to four right angles, the angles AFG, FGC are equal to two right angles. ^^If therefore FB, GD, when the interior angles are equal to two right angles, meet at K, the straight lines FA, GC will also meet if produced; for the angles AFG, CGFaie also equal to two right angles. ''Therefore the straight lines will either meet in both directions or in neither direction, if the two pairs of interior angles are both equal to two right angles. " Let, then, FA, GC meet at Z. "Therefore the straight lines LABK^ LCDK enclose a space : which is impossible. "Therefore it is not possible for two straight lines to meet when the interior angles are e^ual to two right aisles. Therefore they are parallel" [The argument m the words italic^ed would be clearer if it had been shown that the two interior angles on one side of EH are severally equal to the two interior angles on the other, namely BFG to CGF and FGD to AFG\ whence, assuming FB, GD to meet in K, we can take the triangle KFG and place it (e.g. by rotating it in the plane about O the middle point of FG) so diat FG falls where GF\& in the figure and GD fdls on FA, in which case FB must also frdl on GC\ hence, since FB, GD meet at K, GC and FA must meet at a corresponding point Z. Or, as Mr Frankland does, we may substitute for FG a straight line MN through O the middle point of FG drawn peipendicular to one of the parallels, say AB, Then, smce the two triangles OMF, ONG have two angles equal respectively, namely FOM to r z 1. Post. 5] NOTE ON POSTULATE 5 . 205 GON(i. is) and OFMio OGN, and one side O-^ equal to one side OG^ the triangles are congruent, the angle ONG is a right angle, and MN is perpen- dicular to both AB and CD. Then, by the same method of application, MA^ NC are shown to form with MN di triangle MALCN congruent with the triangle NDKBM, and MA^ NC meet at a point L corresponding to K. Thus the two straight lines would meet at the two points K^ JL This is what happens under the Riemann hypothesis, where the axiom that two straight lines cannot enclose a space does not hold, but all straight lines meeting in one point have another point common also, and e.g. in the particular figure just used K^ L are pomts common to all perpendiculars to MN If we suppose that K^ L are not distinct points, but om point, the axiom that two straight lines cannot enclose a space is not contradicted.] II. Ptolemy now tries to prove i. 29 without using our Postulate, and tl.wii deduces the Postulate from it (Proclus, pp. 365, 14 — 367, 27). The argument to prove i. 29 is as follows. The straight line which cuts the parallels must make the sum of the interior angles on the same side equal to, greater than, or less than, two right angles. ^ ^ ? "Let AB, CD be parallel, and let FG meet them. I say (i) that FG does not make the interior angles on the same side greater than two g — right angles. " For, if the angles AFG, CGF are greater than two right angles, the remaining angles BFG, DGF are less than two right angles. *' But the same two angles are also greater than two right angles ; for AF, CG are no more parallel than FB, GD, so that, if the straight line falling on AF, CG makes the interior angles greater than two right angles, the straight line falling on FB, GD will also make the interior angles greater than two right angles. V " But the same angles are also less than two right angles ; for the four angles AFG, CGF, BFG, DGF are equal to four right angles : which is impossible. "Similarly (2) we can show that the straight line falling on the parallels does not make the interior angles on the same side less than two right angles. " But (3), if it makes them neither greater nor less than two right angles, it can only make the interior angles on the same side equal to two right angles.*' III. Ptolemy deduces Post. 5 thus: Suppose that the straight lines making angles with a transversal less than two right angles do not meet on the side on which those angles are. Then, a fortiori, they will not meet on the other side on which are the angles greater than two right angles. Hence the straiglit lines will not meet in either direction ; they are there- fore parallel. But, if so, the angles made by them with the transversal are equal to two right angles, by the preceding proposition (= i. 29). Thaj^fore the same angles will be both equal to and less than two right angles : ▼ which is impossible. Hence the straight lines will meet 9o6 BOOK I [l PdST. 5 IV. Ptolemy lastly enforces his condusion that the strai^t lines will meet on the side on which are the emgUs less than two right am^ by recurring to the ajbrtiofi step in the forgoing proof. Let the angles AFG^ CGJF'm the accompanying figure be together less than two right angles. ' Therefore the angles BFG^ DGF are greater than two right angles. We have proved that the straight lines are not non-secant If they meet, they must meet either towards A^ C or towards B^ D. (i) Suppose they meet towards B^ A at K. Then, smce the angles AFG^ CGFwm less than two right angles, and the angles AFG^ GFB are equal to two right angles, take away the common angle AFG, and the angle CGF is less than the angle BFG; that is^ the exterior angle of the triangle JI^FG is less than the interior and opposite angle BFG : which is impossible. Therefore AB, CD do not meet towards B^ D. (2) But they do meet, and therefore they must meet in one direction or the other : therefore they meet towards A^ B^ that is, on the side where are the angles less than two right angles. The flaw in Rolemy's argument is of course in the part of his proof of I. 29 which I have italicised As Produs says, he is not entitled to iissume that, if AB^ CD are parallel, whatever is true of the interior angles on one side of FG (i.e. that they are together equal to, greater than, or less than, two right angles) is necessarily true at the same time of the interior angles on the ' other side. Ptolemy justifies this by saying that FA^ GC are no more paralld % in one direction tlum FB^ GD are in the other : which is equivalent to the i assumption that through any point only one parallel can be drawn to a given , straight line. That is, he assumes an equivalent of the very Postulate he b : endeavouring to prove. ^ Proclus. I Before passing to his own attempt at a proof, Proclus (p. 368, 26 sqq.) ^ examines an ingenious argument (recalling somewhat the famous one about \ Achilles and the tortoise) which appeared to show that it was impossible for - the lines described in the Postulate to meet j Let AB, CD make with ^C the angles BAC, A CD together less than ^ two right angles. Bisect AC at E and along AB, CD respectively measure AF^ CG so that each is equal to AE. Bisect FG at H and mark off FK, GL each equal to FH\ and so on. Then AF^ CG will not meet at any point on FG ; for, if that were the case, two sides of a triangle would be together equal to the third: which is impossible. e} \ -h fc— 5 r I. Post, s] NOTE OK POSTULATE 5 ao? Simflarly, AB^ CD will not meet at any point on KL \ and ''proceeding like this indefinitely, joining the non-coincident points, bisecting the lines so drawn, and cutting off from the straight lines portions equal to the half of these, they say they thereby prove tlmt the straight lines AB^ CD will not meet anywhere." It is not surprising that Proclus does not succeed in exposing the fallacy here (the fact bemg that the process will indeed be endless, and yet the straight lines will intersect within a finite distance). But Proclus' criticism contains nevertheless something of value. He says that the argument will prove too much, since we have only to join AGva order to see that straight lines making some angles which are together less than two right angles do in fact meet, namely AG^ CO. "Therefore it is not possible to assert, without some definite limitation, that the straight lines produced from angles less than two right angles do not meet On the contrary, it is manifest that same straight lines, when produced from angles less than two right angles, do meet, although the argument seems to require it to be proved that this property belongs to aU such straight lines. For one might say that, the lessening of the two right angles being subject to no limitation, with such and such an amount of lessening the straight lines remain non-secant^ but with an amount of lessening in excess of this they meet (p. 371, 2—10)." [Here then we have the germ of such an idea as that worked out by Lobachewsky, namely that the straight lines issuing from a point in a plane can be divided with reference to a straight line lying in that plane into two classes, ''secant" and "non-secant," and that we may define as parallel the two straight lines which divide the secant from the non-secant class.] Proclus goes on (p. 371, 10) to base his own argument upon "an axiom such as Aristotle too used in arguing that the universe is finite. -For, if from one point two straight lines forming an angle Ife produced indefinitely^ the distance (&acrTao'c¥, Arist. Scoon/fui) between the said straight lines produ^ indefinitely will exceed any finite magnitude. Aristotle at all events showed that, if the straight lines drawn from the centre to the circumference are infinite, the interval between tliem is infinite. For, if it is finite, it is impossible to increase the distance, so that the straight lines (the radii) are not infinite. Hence the straight lines, when produced indefinitely, will be at a distance from one another greater than any assumed finite magnitude." This is a fair representation of Aristotle's argument in De caeh 1. 5, 271 b 28, although of course it is not a proof of what Proclus assumes as an axiom. This being premised, Proclus proceeds (p. 371, 24) : I. " I say that, if any straight line cuts one of two parallels^ it will cut \ the other also, "For let AB, CD be parallel, and let EFG cut AB\ I say that it will cut k CD also. ' " For, since BF^ FG are two straight lines from ^ one point F^ they have, when produced indefinitely, ^ X^ g a distance greater than any magnitude, so that it will ^\ J also be greater than the interval between the parallels. Q ' Whenever therefore they are at a distance from one ^ q another greater than the distance between the parallels, FG wiU cut CD. " Therefore etc" ao8 BOOK I J [l Post. 5 I II. *' Having proved this, we shall prove, as a deduction firom it, the theorem in question. '' For let AB^ CD be two straight lines, and let £^ (idling on them make the angles BEF^ DFE less than two right angles. ''I say that the straight lines will meet on that side on which are the angles less than two right angles. " For, since the angles BEF^ DFE are less than two right angles, let the angle HEB be equal to the excess of two right angles (over them), and let HE be produced to K. ''Since then EF falls on KH^ CD and makes the two interior ang^ HEF^ DFE equal to two right angles, the straight lines HK^ CD are parallel. '' And AB cuts KH\ therefore it will also cut CZ>, by what was before shown. ''Therefore AB^ CD will meet on that side on which are the angles less than two right angles. "Hence the theorem is proved." Clavius criticised this proof on the ground that the asdom from which it starts, taken from Aristotle, itself requires proofl He points out diat, just as you cannot assume that two lines which continually approach one another will meet (witness the hyperbola and its asymptote), so you cannot assume that two lines which continually diverge will ultimately be so fiLr apart that a perpendicular from a point on one let fall on the other will be greater than any assigned distance ; and he refers to the conchoid of Nicororiies, whidi continually approaches its asymptote, and therefore continually gets fartfier away from the tangent at the vertex ; yet the perpendicular from any point on the curve to that tangent will always be less than the distance between the tangent and the asymptote. Saccheri supports the objection. Proclus' first proposition is open to the objection that it assumes that two "parallels" (in the Euclidean sense) or, as we may say, two straight lines which have a common perpendicular, are (not necessarily equidistant, but) so related that, when they are produced indefinitely, the perpendicular from a point of one upon the other remains finite. This last assumption is incorrect on the hyperbolic hypothesis; the "axiom" taken from Aristotle does not hold on the elliptic hypothesis. Na^iraddln at-T^si. The Persian-bom editor of Euclid, whose date is 1 201— 1274, has three lemmas leading up to the final proposition. Their content is substantially as follows, the first lemma being apparently assumed as evident I. (a) If AB^ CD be two straight lines such that successive perpen- diculars, as EF^ GH^ KL^ from points on AB to CD always make with AB unequal angles, which are always acute on the side towards B and always obtuse on Sie side towards A^ then the lines AB^ CD^ so long as they do not cut, approach continually nearer in the direction of the acute angles and diverge continually in the direction of the obtuse angles, and the perpendiculars diminish towards B^ D^ and in- crease towards A^ C. 5 — [^ |!| ( r ip) Conversely, if the perpendiculars so drawn continually become shorter in the direction of B^ Z>, and longer in the I. Post. 5] NOTE ON POSTULATE 5 909 direction of A^ Q the straight lines AB^ CD approach continually nearer in the direction of B^ D and diverge continually in the other direction ; also each perpendicular will make with AB two angles one of which is acute and the other is obtuse, and all the acute angles will lie in the direction towards B^ D^ and the obtuse angles in the opposite direction. [Saccheri points out that even the first part (a) requires proof. As regajrds the converse (p) he asks, why should not the successive acute angles made by the perpendiculars with AB^ while remaining acute, become greater and greater as the perpendiculars become smaller until we arrive at last at a perpendicular which is a common perpendicular to both lines? If that happens, all the author's efforts are in vain. And, if you are to assume the truth of the statement in the lemma without proof, would it not, as Wallis said, be as easy to assume as axiomatic the statement in Post 5 without more ado?] II. ^ AC, BD he drawn from the extremities of AB at right angles to it and on the same side^ and if AC, BD be made equal to one another and CD be joined^ each of the angles ACD, BDC will be rights and CD will be equal to AB. The first part of this lemma is proved by reductio ad absurdum from the preceding lemma. If, e.g., the angle ACD is not right, it must eitiier be acute or obtuse. Suppose it is acute; then, by lemma i, ^^C is greater than BD^ which is contrary to the hypothesis. And so on. The angles ACD^ BDC being proved to be right angles, it is easy to prove that AB^ CD are equal. [It is of course assumed in this "proof" that, if the angle ACD is acute, the angle BDC is obtuse, and vice versa.] III. /n any triangle the three angles are together equal to two right angles. This is proved for a right-angled triangle by means of the forgoing lemma, the four angles of the quadrilateral ABCD of that lemma being all right angles. The proposition is then true for any triangle, since any triangle can be divided into two right-angled triangles. IV. Here we have the final "proof" Of Post. 5, Three cases are distinguished, but it is enough to show the case where one of the interior angles is right and the other acute. Suppose AB^ CD to be two straight lines met by FCE making the angle ECD a right angle and the angle CEB an acute angle. Take any point G on £B^ and draw GJ£ perpendicular to EC Since the angle CEG is acute, the perpendicular GH will fall on the side of E towards 2?, and will either coincide with CD or not coincide with it In the former case the proposition is proved. If GH does not coincide with CD but falls on the side of it towards /^ CZ>, being within the triangle formed by the perpendicular and by C£, EG^ must cut EG. [An axiom is here used, namely that, if CD be produced far enough, it must pass outside the triangle and therefore cut some side, which must be EB^ since it cannot be the perpendicular (i. 27), or CE,] Lastly, let GHtaM on the side of CD towards E. H. B. 14 9IO BOOK I [l Post. 5 Along JTCset of[ HK, KL etc, etch equal to EH^ iintU we get the fint point of division, as M^ beyond C Along GB set off GN^ NO etc, each equal to BG^ until EP ii the mne multiple of EG that ^ATis of EH. llien we can prove that the perpendiculars from N^ O^ P on EC fiUl on the points K^ Z, AT respectively. For take the first perpendicular, that from N^ and call it N& Draw EQ at right angles to Elf and equal to GB^ and set off 5!i? along 5A^also equal to GIf. Join Q(r, GE. Then (second lemma) the angles EQGf QGIfare right, and QG « Elf. Similarly the angles SEG, RGffaxt right, and RG^SH. Thus EGQ is one straight line, and the vertically opposite angles NGR^ EGQ are equal The angles NRG^ EQG are both right, and NG*^ GE^ by construction. Therefore (i. 26) EG = GQ; whence SIf=^ HE « KH^ and 5 coincides with K. We may proceed similarly with tlie other perpendiculars. Thus PM is perpendicular to FE. Hence CD^ being parallel to MP and within the triangle PME^ must cut EP^ if produced f»x enough. John Wallis. As is well known, the argument of Wallis (16x6 — 1703) assumed as a postulate that, given a figure^ another figure is possible which is similar io the gitfen one and of any size whatever. In fiEurt Wdlis assumed this for irbmgles only. He first proved (i) that, if a finite straight line is placed on an infinite straight line, and is then moved in its own direction as far as we please^ it will always lie on the same infinite straight line, (2) that, if an ai^jle be moved so that one 1^ always slides along an infinite straight line^ the angle will remain the same, or equal, (3) that, if two straight lines, cut by a third, make the interior angles on the same side less than two right angles, each of the exterior angles is greater than the opposite interior angle (proved by means of i. 13). (4) If AB, CD make, with A C, the interior angles less than two right angles, suppose AC (with AB rigidly attached to it) to move along AF to the position ay, such that a coincides with C If AB then takes the position aj3, a^ lies entirely outside CD (proved by means of (3) above). (5) With the same hypotheses, the straight line a^, or AB, during its motion^ and before a reaches C, must cut the straight line CD. !6) Here is enunciated the postulate stated above. 7) Postulate 5 is now proved thus. Let AB^ CD be the straight lines which make, with the infinite straight line ACF meeting them, the interior angles BA C, DC A together less than two right angles. Suppose AC (with AB ri^dly attached to it) to move along ACF until AB takes the position of o^ cutting CD in ir. Then, aCW being a triangle, we can, by the above postulate, suppose a triangle drawn on the base CA similar to the triangle aCif. Let it be ACP. [Wallis here interposes a defence of the hypothetical construction.] 1 I. Post. 5] NOTE ON POSTULATE s aii Thus CP and AP meet at P\ and« as by the definition of similar figures the angles of the triangles PCA^ wCa, are respectively equal, the angle PC A being equal to the angle irCa and the angle PAC to the angle waC or BAC^ it follows that CP^ AP lie on CD^ AB produced respectively. Hence AB^ CD meet on the side on which are the angles less than two right angles. [The whole gist of this proof lies in the assumed postulate as to the existence of similar figures ; and« as Saccheri points out, this is equivalent to unconditionally assuming the ''hypothesis of the right angle," and consequently Euclid's Postulate 5.] Gerolamo Saccheri. The book EucUdes etb omni ncuvo vindicaius (1733) by Gerolamo Saccheri (1667 — 1733), a Jesuit, and professor at the University of Pavia, is now accessible (i) edited in German by Engel and Stiickel, DU Theork der ParaUtllinien von Euklid bis auf Gauss^ 1895, PP- 4' — ^3^9 ^"^ (?) ^^ ^^ (Italian version, abridged but annotated, LEudide em€nd€Uo del P, Gerolamo Saccheri^ by G. Boccardini ^Hoepli, Milan, 1904). It is of much greater | importance than all the earlier attempts to prove Post 5 because Saccheri was the first to contemplate the possibility of hypotheses other than that of y Euclid, and to work out a number of consequences of those hypotheses. He was therefore a true precursor of Legendre and of Lobachewsky, as Beltrami called him (1889), and, it might be added, of Riemann also. For, as Veronese observes (Fondamenii di ^ometria^ p. 570), Saccheri obtained a glimpse of the theory of parallels m all its generality, while L^endre, Lobachewsky and G. Bolyai excluded a /r/V^r/, without knowing it, the ''hypo- thesis of the obtuse angle, '^ or the Riemann hypothesis. Saccheri, however, was the victim of the preconceived notion of his time that the sole possible / geometry was the Euclidean, and he presents the curious spectacle of a man , Uiboriously erecting a structure upon new foundations for the very purpose of \ demolishing it afterwards ; he sought for Contradictions in the heart of the I systems which he constructed, in order to prove thereby the falsity of his hypotheses. For the purpose of formulating his hypotheses he takes a plane quadri- lateral ABDCy two opposite sides of wluch, AC^ BD^ are equal and perpendicular to a third AB. Then the angles at C and D are easily proved to be equal On the Euclidean hypothesis they are both right angles; but apart from this hypothesis they might be both obtuse, or both acute. To the three possibilities, which I Saccheri distinguishes by the names (i) the hypothesis of the right angle^ (2) the hypothesis of the obtuse angle and t (3) the hypothesis of the acute angle respectively, there corresponds a certain group of theorems ; and Saccheri's point of view is that the Postulate will be completely proved if the consequences which follow from the last two hypotheses comprise results inconsistent with one another. Amons the most important of his propositions are the following : (i) Ijthe hypothesis of the right angle^ or of the obtuse an^^ or of the acute am^ is proved true in a single case, it is true in ettery other case. (Props, v., VI., Vll.) (a) According as the hypothesis of the right angle, the obtuse angle, or the acute angle is true, the sum of the three .angles of a triangle is equal to, greater than, or less than two right angles. (Prop, ix.) 14—2 C "•-. /^<. A i I fx2 BOOK I [lPoct. 5 (3) From thi exUtence of a sitigk irian^ in tMth ih$ sum of the angUs is equal to^ gnaUr ihan^ or less ikon two right augles tki truth of the hypothesis of the right angle^ obtuse angle^ or acute em^ respeetivefy fottoms. (Pi^ xv.) These propositions involve the foUowing \ If in a single triangle the sum of the angles is equal to, greater than^ or less them two right angUs^ then any triangle has the sum of its angles equal to^ greater thau^ or less than two right angles respectively^ which was (voyed about a century later by Lq;endre for the two cases only where the sum is equal to or less than two right angles. The proofs are not free from imperfections, as when, in the poob of Prop. XII. and the part of Prop. xiii. relating to the hypothesis of the obtuse u angle, Saccheri uses Eucl i. t8 depending on l 16, a proposition which is |( only valid on the assumption that straight lines are infinite tn length ; for this 1 1 assumption itself does not bold under the hypothesis of the obtuse angle (the Riemann hypothesis). The hypothesis of the acute angle takes Saccheri much longer to dispose of, and this part of the book is less satisfactory ; but it contains die following propositions afterwards established anew by Lobachewsky and Bolyai, viz. : (4) Two straight lines in a plane (even on the hypothesis of the acute angle) either have a common perpendieular, or must, if produced in one and the same direction, either intersect once at a finite distance or at least continueUfy approach one another, (Prop, xxiii.) (5) In a cluster of rays issuing from a point there exist aluHxys (on the hypothesis of the acute angle) two determinate straight lines which sej^araie the straight lines which intersect a fixed straight line from those whuh do not . intersect it, ending with and including the straight line which has a common perpendicular with the fixed straight line. (Props, xxx., xxxi., xxxii.) Lambert. A dissertation by G.S. Kliigel, Conatuum praecipuorum theoriam parallelarum demonstrandi recensio (i 763), contained an examination of some thirty '* demon- strations" of Post 5 and is remarkable for its conclusion expressing, apparently for the first time, doubt as to its demonstrability and observing that the \ certainty which we have in us of the truth of the Euclidean hypothesis is « not the result of a series of rigorous deductions but rather of experimental < observations. It also had the greater merit that it called the attention of 1 Johann Heinrich Lambert (1728 — 1777) to the theory of parallels. His Theory of Parallels was vrritten in 1766 and published after his death by G. Bernoulli and C. F. Hindenburg ; it is reprcxluced by Engel and Stackel ! (op. cit. pp. 152 — 208). y The third part of Lambert's tract is devoted to the discussion of the same | three hypotheses as Saccheri's, the hypothesis of the right an^ being for Lambert the first, that of the obtuse angle the second, and that of the acute ] angle the thirds hypothesis; and, with reference to a quadrilateral with three ' right angles from which Lambert starts (that is, one of the halves into which the median divides Saccheri's quadrilateral), Uie three hypotheses are the i assumptions that the fourth angle is a right angle, an obtuse angle, or an ' acute angle respectively. Lambert goes much further than Saccheri in the deduction of new \ propositions from the second and third hypotheses. The roost remarkable is { the following. , The area of a plane triangle^ under the second and third hypotheses, is proportional to the eUfference between the sum of the three cmgla and two right angles. 1 i L Post. $] NOTE ON POSTULATE $ ax3 Thus the numerical expression for the area of a triangle is, under the I Mrd hypothesis t £L^k\w^A^B^C) (i). I and under the sicond hypothesis I A = *(^+i?+C-») (a). ' where ^ is a positive constant A remarkable observation is appended (§ 82) : '' In connexion with this it seems to be remarkable that the second hypothesis holds if spherical instead of plane triangles are taken, because in the former also the sum of the angles is gieater than two right angles, and the excess is proportional to the area of the triangle. ''It appears still more remarkable that what I here assert of spherical triangles can be proved independently of the difficulty of parallels.' This discovery that the second hypothesis is realised on the surface of a sphere is important in view of the development, later, of the Riemann hypothesis (1854). Still more remarkable is the following prophetic sentence: **Iam almost inclined to draw the conclusion that the third hypothesis arises with an imaginary spherical surfau^^ {ci, Lobachewsky's Gkomitrie ima^naire^ 1^37)* ^ No doubt Lambert was confirmed in this by the feurt that, in the formula (2) above, which, for ^ = f', represents the area of a spherical triangle, if r V- I is substituted for r, and r^=^h^ we obtain the formula (i). Legendre. No account of our present subject would be complete without a full reference to what is of permanent value in the investigations of Adrien Marie L^endre (i 752-71833) relating to the theory of parallels, which extended over , the space of a generation. His different attempts to Drove the Euclidean \ hypothesis appeared in the successive editions of his Elkments de Ghmitrie I ^m the first (1794) to the twelfth (1823), which last may be said to contain I his last word on the subject. Later, in 1833, he published, in the Mhnoires de rAcadtmie Royale des Sciences^ xii. p. 367 sqq., a collection of his different proofs under the title Rhflexions sur aijjfirentes mani^res de dhnontrer la thhrie \ des paraUkks. His exposition brought out clearly, as Saccheri had done, and kept steadily in view, the essential connexion between the theory of parallels and the sum of the angles of a triangle. In the first edition of the Elkments the proposition that the sum of the angles of a triangle is equal to two right angUs was proved analytically on the basis of the assumption that the choice of a unit of length does not affect the correctness of the proposition to be proved, which is of course equivalent to Wallis' assumption of the existence of similar figures, A similar analytical proof is given in the notes to the twelfth edition. In his second edition L^endre proved Postulate 5 by means of the assumption that, ^ven three points not in a straight line^ there exists a arcle passing through cUl three. In the third edition (1800) he gave the proposition that the sum of the an^s of a triangle is not greater than two ri^ angles] this proof, which was geometrical, was replaced later by another, the best known, depending on a construction like that of Euclid 1. 16, the continued application of which enables any number of successive triangles to be evolved in which, while the sum of the angles in each remains always equal to the sum of the angles of the original triangle, one of the angles increases and the sum of the other two diminishes continually. But Legendre found the proof of the equally necessary proposition that the sum of the angles of a triangle is JI4 BOOK I [l FdiT. 5 not less than two right angles to present great diflfculties. He first observed that, as in the case of spherical trian^es (in whidi the sum of the angles it greater than. two right angles) the excess of the sum of the apgles over two right angles is proi)ortional to the area of the triangle, so in the case of rectilineal triangles, if the sum of the angles is less than two right angles by « certain defidiy th^ deficit will be pioportbnal to the area of the trian^ Hence if, starting from a given triangle, we could construct another triaiu^ in which the original triangle is contained at least m times, the d^icU of mis new triangle will be equal to at least m times that of the original trian^^ so that the sum of the angles of the greater triang^ will diminish pro g re s s i vdy as m increases, until it becomes lero or negative: which is absurd. The whole difficulty was thus reduced to that of the construction of a trianrie containing the given triangle at least twice; but the solution of even mis simple problem requires it to be assumed (or proved) that tknm^ a ghm /0iftt within a given angle less than two4hirds of a ri^ angle we eon eihoays draw a straight tine which shall meet both sides of the angle. This is however really equivalent to Euclid's Postulate. The proof in the course of whidi the necessity for the assumption appeared is as fbUows. It is required to prove that the sum of the angles of a triai^ cannot be less than two right angles. Suppose A is the least of the three angles of a triangle ABC. Apply to the oj^posite side BC a triangle DBQ eqiud to the triangle ACB, and suoi that the angle DBC is equal to the angle ACB^ and the angle DCB to the angle ABC ; and draw any straight line through D cutting AB^ AC produced in If now the sum of the angles of the triangle f ABC is less than two right angles, being equal to 2^-S say, the sum of the angles of the triangle DBC^ equal to the triangle ABC, is also 2^-S. \ Smce the sum of the three angles of the remaining triangles DEB, FDC I respectively cannot at all events be greater than two right angles [for L^endre's | proofs of this see below], the sum of the twelve angles of tibe four triangles in i the figure cannot be greater than I 4^ + (a^««) + (2/?-«X >e. 8^-28. Now the sum of the three angles at each of the points B^ C, Dv&iR. i Subtracting these nine angles, we have the result that the three angles of ' the triangle AEF cannot be greater than 2^-2$. j Hence, if the sum of the angles of the triangle ABC is less than two right ^ angles by h, the sum of the angles of the laiger triangle AEEis less than two ^ right angles by at least 2S. , We can continue the construction, making a still larger triangle firom AEE, [ and so on. But, however small 8 is, we can arrive at a multiple 2*8 which shall exceed : any given angle and therefore 2B itself; so that the sum of the three angles I of a triangle sufficiently laige would be zero or even less than zero : which is absurd. Therefore etc. The difficulty caused by the necessity of making the above-mentioned assumption made Legendre abandon, in his ninth edition, the inethod of the 1 t I. Post, s] NOTE ON POSTULATE 5 215 editions from the third to the eighth and return to Euclid's method pure and simple. ) But again, in the twelfth, he returned to the plan of constructing any I number of successive triangles such that the sum of the three angles in all of I them remains equal to the sum of the three angles of the original triangle, but two of the angles of the new triangles become smaller and smaller, while the third becomes larger and laiger ; and this time he claims to prove in one proposition that the sum of the three angles of the original triangle is r^a/ to two right angles by continuing the construction of new triangles indefinitely and compressing the two smaller angles of the ultimate triangle into nothing, while the third angle is made to become a flat angle at the same time. The construction and attempted proof are as follows. Let ABC be the given triangle ; let AB be the greatest side and BC the least ; therefore C is the greatest angle and A the least. From A draw AD to the middle point of BC^ and produce AD to C", making AC equal to AB. Produce AB to B^ making AB equal to twice AD. The triangle ABC is then such that the sum of its three angles is equal to the sum of the three angles of the triangle ABC. For take ^ A' along AB equal to AD^ and join CK. Then the triangles ABD^ ACK have two sides and the included angles respectively equal, and are therefore equal in all respects ; and CK is equal to BD or DC Next, in the triangles BCK^ ACD, the angles BKC\ ADC are equal, being respectively supplementary to the equal angles AKC\ ADB\ and the two sides about the equal angles are respectively equal; therefore the triangles BCK^ A CD are equal in all respects. Thus the angle ACB is the sum of two angles respectively equal to the angles B^ C of &e original triangle ; and the angle A in the original triangle is the sum of two angles respectively equal to the angles at A and B in the triangle ABC. It follows that the sum of the three angles of the new triangle ABC is equal to the sum of the angles of the triangle ABC. Moreover, the side AC\ bein^ equal to AB^ and therefore greater than AC^ is greater than BC which is equal to AC. Hence the angle CAB is less than the angle ABC ; so that the angle CAB is less than ^A^ where A denotes the angle CAB of the original triangle. [It will be observed that the triangle ABC is really the same triangle as the triangle AEB obtained by the construction of Eucl. i. 16, but differendy placed so that the longest side k'es along AB.] By taking the midcUe point D of the side B'C and repeating the same construction, we obtain a triangle A B'C' such that (i) the sum of its three angles is equal to the sum of the three angles of ABCf (2) the sum of the I 9i6 BOOK I [lFobt. 5 two angles C'AB\ AB'C* is equal to the aii|^ CAB in tbe preceding triangle, and is therefore less than \A^ and (3) the an|^ CAB* is kas than half &e angle CAB>^ and therefore ie» than ^A. Continuing in this way, we shall obtain a triangle Ahc such that the sum of two angles, those at A and b^ is less than -^A^ and the an|^ at r is greater than the corresponding angle in the preceding triangle. If, L^endre argues, the construction be continued indefinitely so that -^A becomes smaller than any assigned angles the point c ultimately lies on Aby and the sum of the three ai^es of the triangle (which is equal to the sum j of the three angles of the original triangle) becomes identical with the angle | at r, which is then 2^ flat angle, and therdrore equal to two right angles. This proof was however shown to be unsound (in remct Si the final K inference) by J. P. W. Stein in Gergonne's Afmales de MoiUmaHfuis xv., 1824, pp. 77—79- We will now reproduce shortly the substance of the theorems of Legendre which are of the most permanent value as not depending on a particular hypothesis as regards parallels. I. 7%e sum of the thre^ anf^ cf a irian^ cannci be greater than iW0 right atigks. This Legendre proved in two ways. (i) First proof (m the third edition of the iUments). Let ABC be the given triangle, and ACJ9l straight line. Make CE equal to AC, the angle DCE equal to the angle BAC^ and DC equal to AB. Join DE. Then the triangle DCE is equal to the triangle BAC in all respects. If then the sum of the three angles of the triangle ABC is greater than { { 2Ey the said sum must be greater than the sum of the angles BCA^ BCD, DCE, which sum is e^al to 2^. Subtracting the equal angles on both sides, we have the result that i the angle ABC is greater than the angle BCD. ( But the two sides AB, BC of the triangle ABC are respectively equal to i the two sides DC, CB of the triangle BCD. Therefore the base AC is greater than the base BD (EucL i. 24). Next, make the triangle FEG (by the same construction) equal in all ^ respects to the triangle BAC or DCE; and we prove in the same way that * CE (or AC) is greater than DF. And, at the same time, BD is equal to DF^ because the angles BCD, DEFMxe equal. Continuing the construction of fiirther triangles, however small the difference between AC and BD is, we shall ultimately reach some multiple j ' 1.P0ST. s] NOTE ON POSTULATE $ »i7 I of this difference, represented in the figure by (say) the difference between ^. the straight line A/ and the composite line BDFHK^ which will be greater than any assigned length, and greater therefore than the sum of AB vcAJK, Hence, on the assumption that the sum of the angles of the triangle ABC is greater than 2^, the broken line ABDFHKJ may be less than the straight line AJ\ which is impossible. Therefore etc (2) Proof substituted later. If possible^ let 2^ -f a be the sum of the three angles of the triangle ABC^ of which A is not greater than either of the others. Bisect BC at H^ and produce AH to Z>, making HD equal to AH\ join BD. Then the triangles AHC^ DHB are equal in all respects (i. 4) ; and the angles CAH^ ACHzx^ respectively equal to the angles BDH^ DBH. It follows that the sum of the angles of the triangle ABD is equal to the sum of the angles of the original triangle, i.e. to 2^ + a. And one of the angles DAB^ ADB is either equal to or less than half the angle CAB. Continuing the same construction with the triangle ADB^ we find a third triangle in which the sum of the angles is still 2^ + 0, while one of them is equal to or less than ( l CAB)I^. Proceeding in this way, we arrive at a triangle in which the sum of the angles is 2^ + a, and one of them is not greater than ( l CAB)l2\ And, if n is sufficiently large, this will be less than a ; in which case we should have a triangle in which two angles are together greater than two right angles : which is absurd. Therefore a is equal to or less than zero. (It will be noted that in both these proofs, as in Eucl. i. 16, it is taken for granted that a straight One is infinite in length and does not return into itself, which is not true under the Riemann hypothesis.) II. On the assumption that the sum of the angles of a triangle is less than two right angles, if a triangle is made up of two others^ the ^^ deficit** of the former is equal to the sum of the ^^ deficits** of the others. In fact, if the sums of the angles of the component triangles are 2^ - a, i/i-p respectively, the sum of the angles of the whole triangle is (2^-a) + (2-^-/9)-2^ = 2-^-(a + /9). III. If the sum of the three angles of a triangle is equal to two ri^t angles^ the same is true of all trian^es obtained by subdividing it by straight lines drawn from a vertex to meet the opposite side. Since the sum of the angles of the triangle ABC is equal to 2^, if the sum of the andes of the triangle ABD were 2^- a, it would follow that the sum of the angles of the triangle ADC must be 2^ + a, which is absurd (by I. above). IV. If in a triangle the sum of the three angles is equal to two right angles ^ a quadrilcUeral can cUways be constructed with four right angles and four equal sides exceeding in length any assigned rectilineal segment. Let ABC be a triangle in which the sum of the angles is equal to two si8 BOOK I [lFobt. 3 Tight angles. We can assume ABC to be an isaseeies rigfit-am^ki triaiude because we can reduce the case to this by makiiig subdivisions of ABCvr/ straight lines through vertices (as in Prop, iil above). Taking two equal triangles of this kind and placing their hypotennies together, we obtain a quadrilateral with four right angles and four equal sides. Putting four of these quadrilaterals together, we obtain a new quadrilateral of the same kind but with its sides double of those of the first quadrilateral After n such operations we have a quadrilateral with four rigt^t angles and four equal sides, each being equal to s* times the side AB. The diagonal of this qiuidrilateral divides it into two equal isosceles ri^t- angled triangles in each of whicb tibe sum of the angles is equal to two ng^t angles. Consequently, from the existence of one triansle in which the sum of the three angles is equal to two right angles it follows ttiat there exists an isoscetes right-angled triangle widi sides greater than any assigned redDineal segment and such that the sum of its three angles is also equal to two right ang^ V. If the sum of the three angies of one triangle is equal to twoo rigkt angles^ the sum of the three angks of any other triauffe is also ofual to twoo right angies. It is enough to prove this for a rigki-anf/ed triangle, since any triangle can be divided into two right-angled triangles. Let ABC be any right-angled triangle. If then the sum of the angles of any one triangle is equal to two right angles, we can construct (by the preceding Prop.) an isosceles right-angled triangle with the same property and with its perpendicular sides greater than those of ABC Let ABC be such a triangle, and let it be applied to ABC^ as in the figure. Applying then Prop. in. above, we deduce first that the sum of the three angles of the triangle ABC is equal to two right angles, and next, for the same reason, that the sum of the three angles of the original triangle ABC is equal to two right angles. VI. If in any one triangle the sum of the three an^s is less than two right angles, the sum of the three angjUs of any other triangle is also less than two right angles. This follows from the preceding theorem. (It will be observed that the last two theorems are included among those of Saccheri, which contain however in addition the corresponding theorem touching the case where the sum of the angles is greater than two right angles.) We come now to the bearing of these propositions upon Euclid's Postulate 5 ; and the next theorem is VII. If the sum of the three angles of a trian^e is equal to two ri^ angles^ through any point in a plane there can only be drawn one parallel to a givem straight line. I I. Post. 5] NOTE ON POSTULATE 5 319 For the proof of this we require the following Lemma. // is a/ways possible^ through a point P, to draw a straight line which shall mahe^ with a gh^en straight line (r), an angle less than any assigned angle. Let Q be the foot of the perpendicular from i'upon r. Let a segment QR be taken on r, on either side of Q, such that QR is equal to PQ, Join PR^ and mark off the segment RR' equal to PR ; join PR\ If m represents the angle QPR or the angle QRP^ each of the equal angles RPR\ RR'P is not greater than a>/2. Continuing the construction, we obtain, after the requisite number of operations, a triangle /^Rn-i Rn in which each of the equal angles is equal to or less than m/a*. Hence we shall arrive at a straight line PR,^ which, starting from /'and meeting r, makes with r an angle as small as we please. To return now to the Proposition. Draw from P the straight line s perpendicular to PQ. Then any straight line drawn from P which meets r in ^ will form equal angles with r and x, since, by hypothesis, the sum of the angles of the triangle PqR is equal to two right angles. And since, by the I^mma, it is always possible to draw through i'strai^t lines which form with r angles as small as we please, it follows that all the straight lines through P^ except x, will meet r. Hence s is the only parallel to r that can be drawn through P. The history of the attempts to prove Postulate 5 or something equivalent has now been brought down to the parting of the ways. The further developments on lines independent of the Postulate, bc^ning with Schweikart (1780 — 1857), Taurinus (1794 — 1874)^ Gauss (1777 — 1855), Lobachewsky (1793 — 1856), J. Bolyai (1802— 1860) , Riemann (1826— 1866), belong to the history of non-Euclidean geomefry, which is outside the scope of this work. I may refer the reader to the full article Snlla teoria delie parallele e suite geometrie non-euclidee by R. Bonola in Questioni riguardanti I la geometria elementare^ 1900, of which I have made considerable use in the [' above, to the same author's La geometria non-eudidea^ Bologna, 1906, to the first volume of Killing's Einfiihrung in die Grundlagen der Geometrie^ Paderbom, 1893, to P. Mansion's Premiers principes de nUtaghmktrie^ and ) P. Barbann's La gtomttrie non-Euclidienne^ Paris, 1902, to the historical ' summary in Veronese's Fondamenti di geometria^ 1891, p. 565 sqq., and (for original sourc^ to Engel and Stackel, Die Theorie der Parallellinien von Euhlid bis auf Gauss^ 1895, ^^^ Urkunden %ur Geschichie der nicht-Euhlidischen Geometriey i. (Lobachewsky), 1899, and 11. (Wolfgang und Johann Bolyai). I will only add that it was Gauss who first expressed a conviction that the Postulate could never be proved ; this he stated distinctly, first in a review in the Gottingische gdehrte Anseigen, 20 A pril 1816, and secondly in a letter to Bessel of 27 Januaiy, 1829. The actual inaemonstrability of the Postulate was proved by Beltrami (1868) and by Hoiiel {Note sur Timpossibiliti de dSmontrtr par une construetion plane le prindpe de la thkorie des paralUles dit Postulatum d*Euclide in Battaglini's Giomale di matematiche^ viii., 1870, pp. 84 — 89). »o BOOK I [vTon.s Alternatives for Postulate 5. It may be convenient to odiect here a few of the more ndewortby substitutes which have from time to time been fbnnally suggested or tac^ assumed. (i) Through a ghen paini mfy 9m parallel €an he drawn io a given straight line or, Two straight Knes which interseet one another eamnat hath he parallel to one and the same straight Hne. This is commonly known as '' Playfair's Axiom," but it was of coorie not a new discovery. It is distinctly stateo in Produs* note to EucL i. 31. (i a) If a straight line interseei one rf twoo parallels^ it wHl inieruet the other also (Proclus). (i b) Straight lines parallel to the same straight line are paralM to one another. The forms (i a) and (i b) are exactly equivalent to (i). (s) There exist straight Unes everywhirt oguidistemt firmn on€ emctker (Posidonius and Geminus); with which may be compared Produs* ticit assumption that Parallels remain^ throngkout their lenf^n^ at a finite distama from one another. (3) There exists a trian^e in whieh the sum of the three an^ iseqmUto two right angles (L^endre). (4^ Given any figure^ there exists a figure similar to it of any siu wepleau (Walhs, Cftmot, Laplace). Saccheri points out that it is not necessary to assume so much, and that it is enough to postulate that there exist two unequal triemj^ with eqmd angles. (5) Through any point within an angle less than two4hirds of a right at^ a straight line can always be drawn which meets both sides of the anj^e (Legendre). With this may be compared the similar axiom of Lorenz {Grundriss der reinen und angewandten Mathematih^ 1791): Every straight line through a point within an angle must meet one of the sides of the angle, (6) Given any three points not in a straight line^ there exists a circle pcusing through them (Le^ndre, W. Bolyai). (7) . ^* If I could prove that a rectilineal triangle is possible the content of which is greater than any given area^ I cm in a position to prove perfectly rigorousfy the whole of geometry^ (Gauss, in a letter to W. Bolyai, 1799). Cf. the proposition of Legendre numbered iv. above, and the axiom of Worpitzky: TTiere exists no tricmgle in which every angU is cu small as we please. (8) If in a quadrilateral three angles are right angles^ the fourth angle is a right angle also (Clairaut, 1741). (9) If two straight lines are parcUld^ they are figures opposite to (or the reflex of) one another with respect to the middle points of all their transversal segments (Veronese, Elementi^ 1904). Or, Two parallel straight lines intercept, on every transversal which passes through the middle point of a segment included betwun them^ another segment the middle point of which is the middle point of the first (Ingrami, ElemenH, >904). Veronese and Ingrami deduce immediately Playfair's Axiom, ;i NOTES ON THE COMMON NOTIONS »ii AXIOMS OR COMMON NOTIONS. \ ' In a paper Sur Pauthentkiti des axiomes ttEuclide in the Bulletin des sciences nuUMmatiques et ustronamiques^ a* s^r. viii., 1884, p. 162 sqq., Paul ' Tannery maintained that the Common Notions (including the first three) were not in Euclid's work but were interpolated later. The following are his main aiguments. (i) If Euclid had set about distin^ishing between indemon- stmble principles (a) common to all demonstrative sciences and (b) peculiar to geometry, he would, says Tannery, certainly not have placed the common principles second and the special principles (the Postulates) first. (2) If the Common Notions are Euclid's, thb designation of them must be his too ; for he must have used some name to distinguish them from the Postulates and, if he had used another name, such as Axioms^ it is impossible to imagine why that name was changed afterwards for a less suitable one. The word hn^wa {notion\ ^ys Tannery, never signified a notion in the sense of a proposition^ but a notion of some odjat ; nor is it found in any technical sense in Plato and Aristotle. (3) Tannery's own view was that the formulation of the Common Notions dates from the time of Apollonius, and that it was inspired by his work relating to the Elements (we know from Proclus that Apollonius tried to prove the Common Notions), This idea, Tannery thought, was J confirmed by a " fortunate coincidence " furnished by the occurrence of the word Hyyota (notion) in a quotation by Proclus (p. 100, 6): ''we shall agree with Apollonius when he says that we have a nolion (lirocay) of a line when we order the lengths, only, of roads or walls to be measured." In reply to argument (i) that it is an unnatural order to place the purely geometrical Postulates first, and the Common Notions^ which are not peculiar to geometry, last, it may be pointed out that it would surely have been a still more awkward arrangement to give the Definitions first and then to separate from them, by the interposition of the Common Notions^ the Postulates, which are so dosely connected with the Definitions in that they proceed to postulate the existence of certain of the things defined, namely straight lines and circles. (2) Though it is true that wvow, in Plato and Aristotle is generally a notion of an object^ not of difact or proposition, there are instances m Aristotle where it does mean a notion of a fiact : thus in the Eth, Nic, ix. 11, ii7i a32 he speaks of ''the notion (or consciousness) that friends sympathise^ (17 &voia rw (TvraXyciy rovs ^cXovs) and again, b 14, of "the notion (or consciousness) that they are pleased at his good fortune." It is true that Plato and Aristotle do not use the word in a technical sense ; but neither was there apparently in Aristotle's time any fixed technical term for what we call "axioms," since he speaks of them variously as " the so-called axioms in mathematics," " the so- odled common axioms," "the common (things)" (ra xocm), and even "the common opinions " (jcoiml 80^1). I see therefore no reason why Euclid should not himself have given a technical sense to " Common Notions," which is at I ' least a distinct improvement upon "common opinions." (3) The use of hn^wa in Proclus' quotation from Apollonius seems to me .. to be an unfortunate, rather than a fortunate, coincidence from Tannery's point of view, for it is there used precisely in the old sense of the notion of an object (in that case a line). No doubt it is difficult to feel certain that Euclid did himself use the term Common Notions^ seeing that Proclus' commentary generally speaks of Axioms. But even Proclus (p. 194, 8), after explaining the meanmg of the word "axiom," first as used by the Stoics, and secon<fiy as used by "Aristotle and t32 • BOOK I [lCMi the geometers," goes on to say : ''For in their view (that of Aristotk and the geometers) axiom and common naium are the same thing." This, as it seems to me, may be a sort of apology for using the word ''axiom " exdusivdy in what has ^one before, as if Prochis had suddenly bethou^t himself that he had descnbed both Aristotle and the geometers as using the one tenn "axiom," whereas he should have said that Aristotle spoke of "axiomsi* while "the geometers" (in fact EuclidX though meaning the same thing, odled them Common Notions. It may be for a like reason that in another passage (p. 76, 16X after quoting Aristotle's view of an "axiom," as distinct from a portukfr and a hypothesis, he proceeds : " For it is not by virtue of a iommom noHon that, without being taught, we preconceive the circle to be such and such a figure." If this view of the two passages just (quoted is correct, it would strengthen rather than weaken the case for the genwneness of Common NoHoms as the Euclidean term. Again, it is clear from Aristotle's allusions to the "common axioms in mathematics " that more than one axiom of this kind had a place in Uie text- books of his day ; and as he constantly quotes the particular axiom that, y eqnais be taken from equals^ the remainders are equals which is Eudid*^ Common Notion ^ it would seem that at least die first three Common Notions were adopted by Euclid from earlier text-books. It is, besides, scarcely credifa3e that, if the Common Notions which ApoUonius tried to Drove had not been introduced earlier (e.g. by Euclid), they would then have been interpolatad as axioms and not as propositions to be proved. The line taken by Apollonius is much better explained on the assumption that he was directly attacking axioms which he found already admitted mto the Elements. Proclus, who recognised die five Common Notions given in the text, warns us, not only against the error of unnecessarily multiplying the axioms, but against the contrary error of reducing their number unduly (p. 196, 15), "as Heron does in enunciating three only; for it is also an axiom that the whole is greater than the party and mdeed the geometer employs this in many places for his demonstrations, and again that things which coinade are egucU** Thus Heron recognised the first three of the Common Notions ; and this foct, together with Aristotle's allusions to "common axioms*' (in the plural), and in particular to our Common Notion 3, may satisfy us that at least the first three Common Notions were contained in the Elements as they left Euclid's hands. Common Notion i. Ta vf a^rf ura jcal o^Xi/Xotf joriy Zero. Things which are equal to the same thing are also equal to one another, Aristotle throughout emphasises the fact that axioms are self-evident truths, which it is impossible to demonstrate. If, he says, any one should attempt to prove them, it could only be through ignorance. Aristotle therefore would undoubtedly have agreed in Proclus' strictures on Apollonius for attempting to prove the axioms. Proclus gives (p. 194, 25), as a specimen of these attempted proofs by Apollonius, that of the first of the Common Notions. " Let A be equal to Bj and the latter to C; I say that A is also equal to C. For, since A is equal to £/\t A B occupies the same space with it ; and since B is equal to C, it occupies the same space with it. Therefore A also occupies the same space with C" Proclus rightly remarkis (p. 194, 22) that "the middle term is no more . ► I. CM 1—3] NOTES ON COMMON NOTIONS 1—3 iti intelligible (better known, yvwptfuirtfiov) than the conclusion, if it is not actually more disputable." Again (p. 195, 6), the proof assumes two things, (i) that things which "occupy the same space" (roiros) are equal to one another, and (2) that things which occupy the same space with one and the same thing occupy the same space with one another ; which is to explain the obvious by something much more obscure, for space is an entity more unknown to us than the things which exist in space. Aristotle would also have objected to the proof that it is partial and not general (xotfoXov), since it refers only to things which can be supposed to occupy a space (or take up room), whereas the axiom is, as Proclus says (p. 196, i), true of numbers, speeds, and periods of time as well, though of course each science uses axioms in relation to its own subject-matter only. Common Notions 2, 3. 2. Kai iav urocc icra irpo(rr€0j, ra oXa iartv urcu 3. Koi iav airo icrwy ccra dflHup€6i, ^^ KaroXciird/Acva ifrriv icro. 2. If equals be added to equals^ the wholes are equal, 3. If equals be subtracted from equals^ the remainders are equal. These two Common Notions are recognised by Heron and Proclus as genuine. The latter is the axiom which is so favourite an illustration with Aristotle. Following them in the mss. and editions there came four others of the same type as i — ^^3. Three of these are given by Heiberg in brackets ; the fourth he omits altogether. The three are : (a) If equals be adtled to unequals^ the wholes are unequal, iff) Things which are double of the same thing are equal to one another, (c) Things which are halves of the same thing are equal to one anotlur. The fourth, which was placed between {a) and (^), was : (d) If equals be subtracted from unequals^ the remainders are unequal, Proclus, in observing that axioms ought not to be multiplied, indicates that all should be rejected which follow ^m the five admitted by him and appearing in the text above (p. 155). He mentions the second of those just quoted {p) as one of those to be excluded, since it follows from Common Notion I. Proclus does not mention (a), (r) or (d)\ an-NairizI gives (a), {fl\ (b) and (^), in that order, as Euclid's, adding a note of Simplicius that '' three axioms (sententiae acceptae) only are extant in the ancient manuscripts, but the number was increased in the more recent" (a) stands self-condemned because " unequal " tells us nothing. It is easy to see what is wanted if we refer to i. 17, where the same angle is added to a greater and a Uss^ and it is inferred that the first sum is greater than the second. So far however as the wording of (a) is concerned, the addition of equal to greater and less might be supposed to produce less and greater respectively. If therefore such an axiom were given at all, it should be divided into two. Heiberg conjectures that this axiom may have been taken from the commentary of Pappus, who had the axiom about equals added to unequals quoted below {ey^ if so, it can only be an unskilful adaptation of some remark of Pappus, for his axiom {e) has some point, whereas (a) is useless. As regards (b\ I agree with Tannery in seeing no sufficient reason why, if t J i 224 BOOK I [l C.JK we reject it (as we certainly must), the words in i. 47 '^But things irtiich aie double of equals are equal to one another" should be condemned as an interpolation. If they were interpoUted, we should have eipected to find the same interpolation in i. 42, where the axiom is ioMy assumed. I diink it quite possible that Euclid may have inserted such words in one case and left them out in another, without necessarily implying either that he was quoting a formal Common Naium of his own or that he had mi inchided among his Common Notions the particular fiu:t stated as obvious. The corresponding axiom (r) about the kahes of a^uals can hardly be genuine if {b) is not, and Produs does not mention it Tannery acutdy observes however that, when Heibeig, in i. 37, 38, brsckets words statiiw that ^'the halves of equal things are equal to one another** on the ground that axiom {f) was interpolated (althou^ before Theon's time), and explains that Euclid used Common Notion ^ in making his inference, he is clearly mistaken. For, while axiom (b) is an obvious inference from Common Notion s, axiom lA is not an inference from Common Notion 3. Tannery says, in a note, that (q would have to be established by redtutio ad atsunhtm with the help of axiom {b\ that is to say, of Common Notion s. But, as the hypothesis in ttie roAuHo adabsurdum would be that one of the halves ispraier than the other, and it would therefore be necessary to prove that the one whole is gnoUr than the other, while axiom {b) or Common Notion % only refers to oqfuds^ a little argument would be necessary in addition to the refimnce to Common Notion s. I diink Euclid would not have gone through this process in order to prove (^), but would have assumed it as equally obvious with (by Proclus (pp. 197, 6— 19S, 5) definitely rejects two other axioms of the above kind given by Pappus, observing that, as they follow fimn the {pmine axioms, they are rightly omitted in most copies, although Pappus said that they were '' on record " with the others (w¥aj¥aypau^wBai) : (e) If unequais be added to equals^ the difference between the wholes is equal to the difference between the added parts \ and (/) If equals be added to unequais^ the difference between the wholes is equal to the difference between the original unequcUs. Proclus and Simplidus (in an-Nairi2l) give proofs of both. The proof of the former, as given by Simplicius, is as follows : Let A£, CD be equal magnitudes ; and let E£, FD be £ added to them respectively, EB being greater than FD. q I say that AE exceeds CF by the same difference as that by which BE exceeds DF B Cut ofif from BE the magnitude BG equal to DF. Then, since AE exceeds AG by GE^ and AG is equal to CF and BG to DF, AE exceeds CF by the same difference as that by which BE exceeds DF. I Common Notion 4. ^ Kal ra i^apfi6(,oyra cir* oXXi^Aa ura aXXi/Xoif coTiK. * 7%ings whkh coincide with one another are equal to one another. The word c^kpfuSfctv, as a geometrical term, has a different meaning | according as it is used in the active or in the passive. In the passive, | i^apfiHw^ai, it means "to be applied to" without any implication that the app^ed figure will exactly fit, or coincide with, the figure to which it is applied; on the other hand the active i^apfioC€tv is used intransitively and means ''to T r I. C N. 4] NOTES ON COMMON NOTIONS 2—4 ^^5 fit exactly," "to coincide with." In Euclid and Archimedes i^apyuUgmif is constructed with ciri and the accusative, in Pappus with the dative. On Common Notion 4 Tannery observes that it is incontestably geometrical in character, and should therefore have been excluded from the Common Notions; again, it is difficult to see why it is not accompanied by its converse, at all events for straight lines (and, it might be added, angles also), wluch Euclid makes use of in i. 4. As it is, says Tannery, we have here a definition of geometrical equality more or less sufficient, but not a real axiom. It is true that Proclus seems to recognise this Common Notion and the next as proper axioms in the passage (p. 196, 15 — 21) where he says that we should not cut down the axioms to the minimum, as Heron does in giving only three axioms; but the statement seems to rest, not upon authority, but upoo an assumption that Euclid would state explicitly at the b^inning all axioms subsequently used and not reducible to others unquestionably included. Now in I. 4 this Common Notion is not quoted ; it is simply inferred that " the base BC will coincide with EF^ and will be equal to it." The position is therefore the same as it is in regard to the statement in the same propositipn that, *Hf... the base BC does not coincide with EF^ two straight lines will enclose a sface : which is impossible " ; and, if we do not admit that Euclid had the axiom tiiat " two straight lines cannot enclose a space," neither need we infer that he had Common Notion 4. I am therefore inclined to think that the latter is mcn-e likely than not to be an interpolation. It seems clear that the Common Notion, as here formulated, is intended to assert that superposition is a legitimate way of proving the equality of two figures which have the necessary parts respectively equal, or, in other wqrIs, to serve as an axiom of congruence. The phraseology of the propositions, e.g. i. 4 and i. 8, in which Eodid employs the method indicated, leaves no room for doubt that he regarded one figure as actually moved and placed upon the other. Thus in i. 4 he ays, "The triangle ABC being applied (c^op/tofo/Acvov) to the triangle DEF^wcA the point A being plaad {jtM^ivwi) upon the point />, and the straight line AB on DE^ the point B will also coincide with E because AB is equd to DE''\ and in i. 8, " If the sides BA^ AC do not coincide with ED, DF, but fall beside them (take a different position, iropaXXa^ouo-iF), then " etc At the same time, it is clear that Euclid disliked the method and avoided it whereirer he could, e.g. in i. 26, where he proves the equality of two triangles which have two angles respectively equal to two angles and one side of the one equd to the corresponding side of the other. It looks as though he found the method handed down by tradition (we can hardly suppose that, if Thales proved tiiat ) the diameter of a circle divides it into two equal parts, he would do so by any other method than that of superposition), and followed it, in the few cues where he does so, only because he had not been able to see his way to a i satisfactory substitute. But seeing how much of the Elements depends on l 4, directly or indirectly, the method can hardly be regarded as being, in Eodid, of only subordinate importance ; on the contrary, it is fundamental Nor, as < a matter of fact, do we find in the ancient geometers any expression of doubt as to the legitimacy of the method. Archimedes uses it to prove that any . spheroidal figure cut by a plane through the centre is divided into two equal I parts in respect of both its surface and its volume; he also postulates in Equilibrium of Planes i. that " when equal and similar plane figures coincide if applied to one another, their centres of gravity coincide also." j Killing (Einfuhrung in die Grundlagen der Geometrie^ 11. pp. 4, 5) H. E. 15 236 BOOK I [lCMa contrasts the attitude of the Greek geometers with that of the philosc^riieni who, he says, appear to have agreed in banishing motion from geometry altogether. In support of this he refers to the view frequently expressed l^ Aristotle that mathematics has to do with immovable objects (aiciri|raX and that only where astronomy is admitted as part of mathematical science is motion mentioned as a subject for mathematics. Cf. MiU^h. 989 b 32 ''For madie- matical objects are among things which exist i^Mut from motion, except sudi as relate to astronomy"; Metapk. 1064 a 30 ''Physics deals with things which have in themselves the principle of motion; mathematics is a theoretical science and one concerned with things which are siatumary (/imirra) but not separable" (sc. from matter^; in Physics 11. 2, 193 b 34 he speaks of the subjects of mathematics as "m thought separable from motion." But I doubt whether in Aristotle's use of the words "immovaUe," "widi- out motion" etc as applied to the subjects of mathematics diere is any implication such as Killing supposes. We arrive at mathematical concepts by abstraction from material objects ; and just as we, in thought, eliminate the matter, so according to Aristotle we diminate the attributes of matter as such, e.g. qualitative clmnge and motion. It does not appear jto me that die use of " immovable " in the passages referred to means more than this. I do not think that Aristotle would have r^purded it as illegitimate to mom z. geometrical figure from one position to another; and I infer this from a passage in De caelo iii. i where he is criticising "those who make up every iKxly that has an origin by putting together pianes^ and resolve it again into planesJ' The reference must be to the TimaeHS (54 B sqq.) where Plato evolves the four elements in this way. He begins with a right-angled triangle in which the hypotenuse is double of the smaller side; six of these put together in the proper way produce one equilateral triangle: Making solid uigles with (a) three, (b) four, and (c) five of these equilat^al triangles respectively, and taking the requisite number of these solid angles, namely four of (tf), six of (b) and twelve of (c) respectively, and putting them together so as to form regular solids, he obtains (a) a tetrahedron, (fi) an octahedron, (y) an icosahedron respectively. For the fourth element (earth), four isosceles right-angled triangles are first put together so as to form a square, and then six of these squares are put together to form a cube. Now, says Aristotle (299 b 23), '*it is absurd that planes should only admit of being put together so as to touch in a line', for just as a line and a line are put together in both ways, lengthwise and breadthwise, r so must a plane and a plane. A line can be combined with a line in the sense of being a line superposed^ and not adde(P^\ the inference being that deplane can | be superposed on biplane. Now this is precisely the sort of motion in question ! here; and Aristotle, so far from denying its permissibility, seems to blame ^ Plato for not using it. Cf. also Physics v. 4, 228 b 25, where Aristotle speaks ' of *' the spiral or other magnitude in which any part will not coincide with , any other part," and where superposition is obviously contemplated. Motion without deformation. It is well known that Helmholtz maintained that geometry requires us to assume the actual existence of rigid bodies and their free mobility in space, . whence he inferred that geometry is dependent on mechanics. Veronese exposed the fallacy in this (Fondamenti di geometria^ pp. xxxv — ^Dovi, 239 — 240 note, 615 — 7), his argument being as follows. . Since geometry is concerned with empty space, which is immovable, it would be at least strange if it was necessary to have recourse to the real motion of bodies for a definition. i I UC.M4] NOTE ON COMMON NOTION 4 227 and for the proof of the properties, of immovable space. We must distinguish \ the intuitive principle of motion in itself from that of motion without deforma- tion. Every point of a figure which moves is transferred to another point in space. " Without deformation " means that the mutual relations between the points of the figure do not change, but the relations between them and other figures do change (for if they did not, the figure could not move). Now consider what we mean by saying that, when the figure A has moved from the position Ax to the position ^,, the relations between the points of A in the position A^ are unaltered from what they were in the position A^^ are the same in fact as if ^ had not moved but remained at A^^ We can only say that, judging of the figure (or the body with its physical qualities eliminated) by the impressions it produces in us during its movement, the impressions produced in us in the two different positions (which are in tim^ distinct) are equal. In fact, we are making use of the notion of equality between two distinct figures. Thus, if we say that two bodies are equal when they can be superposed by means of movement without deformation^ we are com- mitting di petitio principii. The notion of the equality of spaces is really prior to that of rigid bodies or of motion without deformatioa Helmholtz supported his view by reference to the process of measurement in which the measure must be, at least approximately, a rigid body, but the exbtence of a rigid body as a standard to measure by, and the question how we discover two equal spaces to be equal, are matters of no concern to the geometer. The method of superposition, depending on motion without deformation, is only of use as 9i practical test \ it has nothing to do with the theory of geometry. Compare an acute observation of Schopenhauer (Die Welt als WilU^ 2 ed. 1844^ II. p. 130) which was a criticism in advance of Helmholtz' theory : "I am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the eighth is not rather attacked : ' Figures which coincide (sich decken) are equal to one another.' For coincidence (das Sichdecken) is either mere tautology, or something entirely empirical, which belongs, not to pure intuition (Anschauung), but to external sensuous experience. It presupposes in fact the mobility of figures; but that which is movable in space is matter and nothing else. Thus this appeal to coincidence means leaving pure space, the sole element of geometry, in order to pass over to the material and empirical." Mr Bertrand Russell observes (En^clopaedia BritannicOy Suppl. Vol. 4, 1902, Art '* Geometry, non-Euclidean ") that the apparent use of motion here is deceptive ; what in geometry is called a motion is merely the transference of our attention from one figure to another. Actual superposition, which is nominally employed by Euclid, is not required; all that is required is the transference of our attention from the original figure to a new one defined by the position of some of its elements and by certain properties which it shares with the original figure. If the method of superposition is given up as a means of defining theoreti- cally the equab'ty of two figures, some other definition of equality is necessary. But such a definition can be evolved out of empirical or practical observation of the result of superposing two material representations of figures. This is done by Veronese (Ekmenti di geometria^ 1904) and Ingrami (Elementi di geometria, 1904). Ingrami says, namely (p. 66): " If a sheet of paper be folded double, and a triangle be drawn upon it and then cut out, we obtain two triangles superposed which we in practice call equal. If points ^, J?, C, Z> ... be marked on one of the triangles, then, when we place this triangle upon the other (so as to coincide with it), we see IS— 2 228 BOOK I [l C. N. 4 that each of the particular points taken on the first is superposed on one particular point of the second in such a way that the segments AB^ AC, AD, BQ BD^ CDy ... are respectively superix>sed on as many segments in die second triangle and are therefore equal to them respectively. In this way we justify the following '' Definition of equality. ''Any two figures whatever will be called equal when to the pdnts of one the points of the other can be made to correspond univacalh [Le. every 0me point in one to one distinct point in the other and viu vena] in such a way that the segments which join the points, two and two, in one figure are respectively equal to the segments which join, two and two, the corresponding pomts in the other." Ingrami has of course previously postulated as known the signification of the phrase equa/ {rectilineal) segments, of which we get 9i practical notion when we can place one upon the other or can place a third movable segment successively on both. New systems of Congruence-Postulates. In the third Article of Questiom riguardanti la geametria ekmentart, ipop, pp. 65 — 82, a review is given of three difierent systems: U) that of Pas(£ in Vorl^ngen Oder neuere Geamdrk, 1882, p. loi sqq., (2) that of Veronese according to the FondammH H geametria, 1891, md die ElemenH taken together, (3) that of Hilbert (see Chundlagen der Geametrie, 1903, pp. 7 — 15). These systems difier in the particular conceptions taken by the three authors as primary, (i) Pasch considers as primary the notion of congruence or equality between any figures which are made up of a finite number of points only. The definitions of congruent segments and of congruent angles have to be deduced in the way shown on pp. 68 — 9 of the Article refened to, after which EucL i. 4 follows immediately, and Eucl. i. 26 (i) and i. 8 by a method recalling that in Eucl. i. 7, 8. (2) Veronese takes as primary the conception of congruence between segments (rectilineal). The transition to congruent angles^ and thence to triangles is made by means of the following postulate : ''Let AB^ AC and A'B\ AC be two pairs of straight lines intersecting at A^ A\ and let there be determined upon them the congruent segments AB, AB* and the congruent segments AC, AC\ then, if BC^ B*C are congruent, the two pairs of straight Una are con- gruent" (3) Hilbert takes as primary the notions of congruence between twth segments and angles, . It is observed in the Article referred to that, from the theoretical stand- [ point, Veronese's system is an advance upon that of Pasch, since the idea of congruence between segments is more simple than that of congruence between t any figures ; but, didactically, the development of the theory is more compli- ' cated when we start from Veronese's system than when we start from that of \ Pasch. The system of Hilbert offers advantages over both the others from the point of view of the teaching of geometry, and I shall therefore give a short account of his system only, following the Article above quoted. i I. C, N. 4] NOTE ON COMMON NOTION 4 229 Hubert's system. The following are substantially the Postulates laid down. (i) If one s^mmt is congruent ivith another^ the second is also congruent with the first, (2) If an angle is congruent with another angU^ the second angle is also congruent with the first, (3) Two segments congruent with a third are congruent with one another, (4) 7\vo angles congruent with a third are congruent with one another, (5) Any segment AB is congruent with itself^ independently of its sense. This we may express symbolically thus : AB^AB^BA, (6) Any angle (ab) is congruent with itself independently of its sense. This we may express symbolically thus : (ab) = {ab) = (ba), (7) On any straight line r*, starting from any one of its points A', and on each side of it respectively^ there exists one and only one segment congruent with a segmeftt AB belonging to the straight line r. (8) Given a ray a, issuing from a point O, in any plane which contains it and on each of the two sides of it^ there exists one and only one ray b issuing from O stich that the angle (ab) is congruent with a given angle (a'b'). (9) ^ AB, BC are two consecutive segments of the same straight line r {segments^ that is, having an extremity and no other point common), and A'B', B'C two consecutive segments on another straight line r', and if AB = A'B', BC s B'C, then AC^AC, I (10) j^ (ab), (be) are two consecutiife angles in the same plane ir (angles, (that is, having the ifertex and one side common), and (a'b'), (b'c') two consecu- tive angles in another plane v, and if (A) = (a'b'), (be) = (b'c'), then ^ (11) If two trian^s have two sides and the included angles respectively congruent, they have also their third sides congruent as well as the angles opposite to the congruent sides respectiwly, I As a matter of fact, Hilbert's postulate corresponding to ^11) does not ( assert the equality of the third sides in each, but only the equality of the two remaining angles in one triangle to the two remaining angles in the other respectively. He proves the equality of the third sides (thereby completing the theorem of Eucl. i. 4) by reductio ad absurdum thus. ljtiABC,AB'C' A A' be the two triangles which have the ^^ ^y^^ sides AB, AC respectively congruent ^/^ \ ^/^ A with the sides AB', A'C and the ^^ \ y^ \ \ included angle at A congruent with ^ ^ ^7- g-jj. the included anple at A\ Then, by Hilbert's own postulate, the angles ABC, ABC are congruent, as also the angles ACB, AC'B, If BC is not congruent with BC, let D be taken on BC such that BC, BD are congruent, and join AD, 7 930 BOOK I [i. C.N.A Then the two triangles ABQ ASD have two rides and the included angles congraent respectively; therefore, by the same postulate^ the angles BACy BAD are congruent But the angles BAC^ BA'C are congruent ; therefore, by (4) above, the angles BAC^ BAD are congruent : which is impossible, since it contradicts (8) above. Hence BC^ BC cannot but be congruent Eucl. I. 4 is thus proved \ but it seems to be as well to indude all of that theorem in the postulate, as b done in ^i i) above, rince the two parts of it are equally suggested by empirical observation of the result of one superporition. A proof similar to that just given immediately establishes Eud. i. 26 (i), and Hilbert next proves that If two angles ABC, A'B'C are eangnteni with one another^ their supplt- mentary angles CBD, CB'D' are also eo^pnent with one another. We choose Ay D on one of the straight lines forming the first angle, and A\ D on one of those forming the second angle, and agam C, C on the other O C /^^^^^ straight lines forming the angles, so that AB is congruent with AB^ C*B with CB, and DB with DB. The triangles ABC, ABC are congruent, by (11) above; and AC is congruent with AC, and the angle CAB with the angle CAB. Thus, AD, AD being congruent, by (9), the triangles CAD, CAB are also congruent by (11); / whence CD is congruent with CB, and the angle ADC with the angle ADC. Lastly, by (11), the triangles CDB, CDB are congruent, and the angles CBD, CBD are thus congruent Hilbert's next proposition is that | Given that the angle (h, k) in the plane a is congruent with the angle (h', k') I in the plane a, and that 1 is a half-ray in the plane a starting from the vertex | of the angle (h, k) and lying within that angle, there always exists a half-ray V [ in the second plane a, starting from the vertejC of the an^ (h', k') and lying i within that an^ such that y (h,i)=(h'.i'). ««/(k.i)=(k',i'). • If O, O are the vertices, we choose points A, B on h, h, and points A, B t on K, K respectively, such that OA, OA are congruent and also OB, OB: * \ ( The triangles OAB, OAB are then congruent ; and, if / meets AB in C, we can determine C on AB such that ^'C is congruent with AC. Then f drawn from O through C is the half-ray required. T" I. C N. 4] NOTE ON COMMON NOTION 4 a3i The congruence of the angles (^ /), {h\ f) follows from (11) directly, and that of (k, i) and (^, f) follows in the same way after we have inferred by means of (9) that, AB, AC being respectively congruent with A'F, A*C\ the difference BC is congruent with the difference ffC\ It is by means of the two propositions just given that Hilbert proves that All right angles are congruent with one another. Let the angle BAD be congruent with its adjacent angle CAD^ and Ukewise the angle BAU congruent with its adjacent angle CA'Jff. All four angles are then right angles. O'T QW D' If the angle BA'D is not congruent with the angle BAD, let the angle with AB for one side and congruent with the angle BA*D be the angle BAD\ so that AD' falls either within the angle BAD or within the angle DAC. Suppose the former. By the last proposition but one (about adjacent angles), the angles BA D^ BAD* being congruent, the angles CA*D, CAD' are congruent Hence, by the hypothesis and postulate (4) above, the angles BAD\ CAD* are also congruent And, since the angles BAD, CAD are congruent, we can find within the angle CAD a half-ray CAD'* such that the angles BAD\ CAD'' are congruent, and likewise the angles DAD\ DAD" (by the last proposition). But the angles BAD\ CAD' were congruent (see above); and it follows, by (4), that the angles CAD'^ CAD" are congruent: which is impossible, since it contradicts postulate (8). Therefore etc Euclid I. 5 follows directly by applying the postulate (11) above to ABC, A CB as distinct triangles. Postulates (9), (10) above give in substance the proposition that "the sums or differences of s^;ments, or of angles, respectively equal, are eqxiaL" Lastly, Hilbert proves EucL i. 8 by means of the theorem of Eud. i. 5 and the proposition just stated as applied to angles. ABC, A'ffC being the given triangles with three sides respectively congruent, we suppose an angle CBA" to be deter- mined, on the side of BC opposite to A, congruent with the angle A'BC, and we make BA" equal to A'B, The proof is obvious, being equivalent to the alternative proof often given in our text-books for Eud. i. 8. 232 BOOK I [i. CN.S Common Notion 5. fcoi ri i\o¥ rw fiifHw^ fi€l(fi¥ [Ivrtr]. 7^ whole is greater than the pari. Proclus includes this "axiom " on the same ground as the preceding one. I think however there is force in the objection which Tannery takes to it, namely that it replaces a difertni expression in EudL i. 6, where it is stated that "the triangle DBC wdl be equal to the triangle ACB^ the ks$ to the greater: which is absurd!* The axiom appears to be an abstraction or generalisation substituted for an immediate inference from a geometrical figure, but it takes the form of a sort of definition of whole and part The probabilities seem to be against its being genuine, notwithstanding Proclus* approval of it Qavius added the axiom that the whok is the equal to the sum of its parts. Other Axioms introduced after Euclid's time. [9] Ikuo straight lines do not enclose (or eoniain) a space. Proclus (g. 196, 21) mentions this in illustration of the undue multiplication of axioms, and he points out, as an objection to it, that it belongs to the subject matter of geometry, whereas axioms are of a general character, and not peculiar to any one science. The real objection to the axiom is that it is unnecessary, since the fact which it states is included in the meanina; of Postulate I. It was no doubt taken firom the passage in i. 4, ''if...the oase BC does not coincide with EF^ two straight lines wul enclose a space-, which is impossible** \ and we must certainly re^ud it as an interpolation, notwith- standing that two of the best iiss. have it after Postulate 5, and one gives it as Common Notion 9. Pappus added some others which Proclus objects to (p. 198, 5) because they are either anticipated in the definitions or follow firom them. te) ' All the parts of a plane^ or of a straight line, coincide with one another. (h) A point divides a line^ a line a surface, and a surf cue a solid \ on which Proclus remarks that everything is divided by the same things as those by which it is bounded. An-Nairizi (ed. Besthom-Heibcrg, p. 31, ed. Curtze, p. 38) in his version of this axiom, which he also attributes to Pappus, omits the reference to solids, but mentions planes as a particular case of surfaces. " (a^ A surface cuts a surface in a line ; (P) If two surfaces which cut one another are plane, they cut one another in a straight line ; (y) A line cuts a line in a point (this last we need in the first proposition)." ^ {h) Magnitudes are susceptible of the infinite (or unlimited) both by way of addition and by way of successive diminution, but in both cases potentially only [ M airttfioy iv rocs iirfiOtfriv iartv jcou rg irpo<rO€(r€i jccu rg iinKa$<up4a'€L, Swofici Sk heirtpov). i An-Nairizl's version of this refers to straight lines and plane surfaces only : ' "at regards the straight line and the plane surface, in consequence of their . evenness, it is possible to produce them indefinitely. This "axiom" of Pappus, as quoted by Proclus, seems to be taken directly I firom the discussion of ro an-cipov in Aristotle, Physia in. 5 — 8, even to the wording, for, while Aristotle uses the term division (8ia(Jpc(ric) most frequently as the antithesis of addition (<rw0c(ricV he occasionally speaks of subtraction (d^aipco-if) and diminution (Ka^oipco-ic). Hankel {Zur Geschichte der Mathe- matih im Alterthum und MittelaUer, 1874, pp. 119 — 120) gave an admirable f \ 1. Axx.] ADDITIONAL AXIOMS 233 summary of Aristotle's views on this subject ; and they are stated in greater detail in Gorland, AristoteUs und die Mathematik^ Marburg, 1899, PP* ^57 — 183. The infinite or unlimited (dirc4f>ov) only exists potentially (8wa/ict), not in actuality (^cpycti^). The infinite is so in virtue of its endlessly changing into something else, like day or the Olympic Games {Phys, in. 6, 206 a 15 — 25). The infinite is manifested in different forms in time, in Man, and in the division of magnitudes. For, in general, the infinite consists in something new being continually taken, that something being itself always finite but always different. Therefore the infinite must not be regarded as a particular thing (to3c ri), as man, house, but as being always in course of becoming or decay, and, though finite at any moment, always different from moment to moment. But there is the distinction between the forms above referred to that, whereas in the case of magnitudes what is once taken remains, in the case of time and Man it passes or is destroyed but the succession is unbroken. The case of addition is in a sense the same as that of division ; in the finite magnitude the former takes place in the converse way to the latter ; for, as we see the finite magnitude divided ad infinitum^ so we shall find that addition gives a sum tending to a definite limit I mean that, in the case of a finite magnitude, you may take a definite fraction of it and add to it (continually) in the same ratio ; if now the successive added terms do not include one and the same magnitude whatever it is [i.e. if the successive terms diminish in geometrical progression], you will not come to the end of the finite magnitude, but, if the ratio is increased so that each term does include one and the same magnitude whatever it is, you will come to the end of the finite magnitude, for every finite magnitude is exhausted by continuaUy taking from it any definite fraction whatever. Thus in no other sense does the infinite exist, but only in the sense just mentioned, that is, potentially and by way of diminution (206 a 25 — b 13). And in this sense you may have potentially infinite addition^ the process being, as we say, in a manner, the same as with division ad infinitum : for in the case of addition you will always be able to find some- thing outside the total for the time being, but the total will never exceed every definite (or assigned) magnitude in the way that, in the direction of division, the result will pass every definite magnitude, that is, by becoming smaller than it The infinite therefore cannot exist even potentially in the sense of exceeding every finite magnitude as the result of successive addition (206 b 16 — 22). It follows that the correct view of the infinite is the opposite of that commonly held : it is not that which has nothing outside it, but that which always has something outside it (206 b 33 — 207 a i). Contrasting the case of number and magnitude, Aristotle points out that (i) in number there is a limit in the direction of smallness, namely unity, but none in the other direction : a number may exceed any assigned number however great ; but (2) with magnitude the contrary is the case : you can find a magnitude smaller than any assigned magnitude, but in the other direction there is no such thing as an infinite magnitude (207 b r-^5). The latter assertion he justified by the following argument However large a thin^ can be potentially, it can be as large actually. But there is no magnitude perceptible to sense that is infinite. Therefore excess over every assigned magmtude is an impossibility; otherwise there would be something lu^er than the universe (o^yos) (207 b 17 — 21). Aristotle is aware that it is essentially of physical magnitudes that he is speaking. He had observed in an earlier passage {Phys, in. 5, 204 a 34) that it is peiiiaps a more general inquiry that would be necessary to determine I 234 BOOK I [i. Axx. whether the infinite is possible in mathematics, and in the domain of thought and of things which have no magnitude; but he excuses himself from entenng upon this inquiry on the ground that his subject is physics and sensible objects. He returns however to the bearing of his conclusions on mathematics in III. 7, 207 b 27 : "my argument does not even rob mathematicians of their study, although it denies the existence of the infinite in the sense of actual existence as something increased to such an extent that it cannot be gone through (dSic^tryrov) ; for, as it is, thc^ do not even need the infinite or use it, but only require that the finite (straight line) shall be as long as they please; and another magnitude of any size whatever can be cut in the same ratio as the greatest magnitude. Hence it will make no difference to them for the purpose of demonstration." Lastly, if it should be urged that the infinite exists in thought^ Aristotle replies that this does not involve its existence in fact, A thing is not greater than a certain size because it is conceived to be so, but because it i>; and magnitude is not infinite in virtue of increase in thought (208 a 16 — 22). Hankel and Gorland do not quote the passage about an infinite series of magnitudes (206 b 3 — 13) included in the above paraphrase; but I have thought that mathematicians would be interested in the distinct e3q>ression of Aristotle's view that the existence of an infinite series the terms of which are magnitudes is impossible unless it is convergent, and (with reference to Riemann's developments) in the statement that it does not matter to geometiy if the straight line is not infinite in length, provided that it is as long as we please. Aristotle's denial of even the potential existence of a sum of magnitudes which shall exceed every definite magnitude was, as he himself implies, in conflict with the lemma or assumption used by Eudoxus (as we infer firom Archimedes) to prove the theorem about the volume of a pyramid. The lemma is thus stated by Archimedes (Quadrature of a parabola^ preface): " The excess by which the greater of two unequal areas exceeds the less can, if it be continually added to itself, be made to exceed any assigned finite area." We can therefore well understand why, a century later, Archimedes felt it necessary to justify his own use of the lemma as he does in the same preface : " The earlier geometers too have used this lemma : for it is by its help that they have proved that circles have to one another the duplicate ratio of their diameters, that spheres have to one another the triplicate ratio of their diameters, and so on. And, in the result, each of the said theorems has been accepted no less than those proved without the aid of this lemma." Principle of continuity. The use of actual construction as a method of proving the existence of J figures having certain properties is one of the characteristics of the Elements. Now constructions are effected by means of straight lines and circles drawn I in accordance with Postulates i — 3 ; the essence of them is that such straight ^ lines and circles determine by their intersections other points in addition to \ those given, and these points again are used to determine new lines, and so on. This being so, the existence of such points of intersection must be postulated ) or proved in iht same way as that of the lines which determine them. Yet ] there is no postulate of this character expressed in Euclid except Post. 5. / This postulate asserts that two straight lines meet if they satisfy a certain condition. The condition is of the nature of a &opiirfio« (discrimination^ or condition of possibility) in a problem ; and, if the existence of the point of r I.AXX.] PRINCIPLE OF CONTINUITY 235 intersection were not granted, the solutions of problems in which the points of intersection of straight lines are used would not in general furnish the required proofs of the existence of the figures to be constructed. But, equally with the intersections of straight lines, the intersections of circle with straight line, and of circle with circle, are used in constructions. Hence, in addition to Postulate 5, we require postulates asserting the actual existence of points of intersection of circle with straight line and of circle with circle. In the very first proposition the vertex of the required equilateral triangle is determined as one of the intersections of two circles, and we need therefore to be assured that the circles will intersect Euc lid seems to assume U ita s obvious, although it is not so ; and he makes a similar assumption m XTii, It is true that in the latter case Euclid adds to the enunciation that two of the given straight lines must be together greater than the third ; but there is nothing to show that, if this condition is satisfied, the construction is always possible. In i. 12, in order to be sure that the circle with a given centre will intersect a given straight line, Euclid makes the circle pass through a point on the side of the line opposite to that where the centre is. It appears therefore as if, in this case, he based his inference in some way upon the , definition of a circle combined with the fact that the point within it called the centre is on one side of the straight line and one point of the circumference on the other, and, in the case of two intersecting circles, upon similar con- siderations. But not even in Book in., where there are several propositions about the relative positions of two circles, do we find any discussion of the conditions under which two circles have two, one, or no point common. The deficiency can only be made good by the Principle of Continuity, Killing {Einfiihrung in die Grundi^gen der Geometrie^ 11. p. 43) gives the following forms as sufficient for most purposes. (a) Suppose a line belongs entirely to a figure which is divided into two parts ; then, if the line has at least one point common with each part, it must also meet the boundary between the parts; or (p) If a point moves in a figure which is divided into two parts, and if it belongs at the beginning of the motion to one part and at the end of the motion to the other part, it must during the motion arrive at the boundary between the two parts. In the Questioni riguardanti la geometria elementare^ Article 4 (pp. 83 — loi), the principle of continuity is discussed with special reference to the Postulate of Dedekmd, and it is shown, first, how the Postulate may be led up to and, secondly, how it may be applied for the purposes of elementary geometry. Suppose that in a segment AB of a straight line a point C determines two segments A C, CB, If we consider the point C as belonging to only one of the two segments ACy CB^ we have a division of the segment AB into two parts with the following properties. 1. Every point of the segment AB belongs to one of the two parts. 2. The point A belongs to one of the two parts (which we will call the firsi) and the point B to the other ; the point C may belong indifferently to one or the other of the two parts according as we choose to premise. 3. Every point of the first part precedes every point of the second in the order AB of the segment (For generality we may also suppose the case in which the point C fedls at A or at B. Considering C, in these cases respectively, as belonging to the first or the second part, we still have a division into parts which have the properties above enunciated, one part being then a single point A or B.) L. 236 BOOK I [lAxx. Now, considering carefully the inverse of the above proposition, we see that it agrees with the idea which we have of the continuity of the straight line. Consequently we are induced to admit as 9,poshdaU the following. If a segment of a straight line AB is divided into two parts so that {iS every point of the segment AB bdongs to one of theparts^ (2) t?u extremity A belongs to the first part and "^ to the second^ and (3) any point whatever of the first part precedes any point whatever ef ike seamd party in the order AB of the segment^ there exists a point C of the segment AB (which may belong either to ame part or to the other) such that every point of AB that precedes C belongs to the first party and every point of AB that follows C belongs to the second part in the division originally assumed (If one of the two parts consists of the single point A or B^ the point C is the said extremity A ox B oi the segment) This is the Postulate of Dedekind, which was enunciated by Dedekind himself in the following slightly different form (Stetigheit und irrationale ZcMen^ 1872, new edition 1905, p. 11). '' If all points of a straight line fall into two classes such that every paint of the first class lies to the left of every point of the second class^ there exists one emd only one point which produces this division of all the points into two classes^ this division of the straight line into two parts,** The above enunciation may be said to correspond to the intuitive notion which we have that, if in a segment of a straight line two points start from the ends and describe the segment in opposite senses, they meet in a point The point of meeting might be regarded as belonging to both parts, but for the present purpose we must regard it as belonging to one only and subtracted from the other part Application of DedehincPs postulate to angles. If we consider an angle less than two right angles bounded by two rays a, by and draw the straight line connecting Ay a point on a, with By a point 4 on by we see that all points on the finite s^ment AB correspond univocally to all the rays of the angle, the point corresponding to any ray being the point in which the ray cuts the segment AB \ and if a ray be supposed to move about the vertex of the angle from the position a to the position by the corresponding points of the segment AB are seen to follow in the same order as the corresponding rays of the angle {ab). Consequently, if the angle (ab) is divided into two parts so that (i) each ray of the angle (ab) belongs to one of the two parts, it) the outside ray a belongs to the first part and the ray b to the second, (3) any ray whatever of the first part precedes any ray whatever of the ] second part, i the correspondmg points of the segment AB determine two parts of the i segments such that \ ii) every point of the segment AB belongs to one of the two parts, ) 2^ the extremity A belongs to the first part and B to the second, 3) any point whatever of the first part precedes any point whatever of '; the second. I But in that case there exists a point C of AB (which may belong to one or the other of the two parts) such that every point of AB that precedes C belongs to the first part and every point of AB that follows C belongs to the second part I s I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 237 Thus exactly the same thing holds of Cy the ray corresponding to C, with reference to the division of the angle (ab) into two parts. It is not difficult to extend this to an angle (ab) which is either flat or greater than two right angles ; this is done (Bonola, op, cit. pp. 87 — 88) by supposing the angle to be divided into two, (ad)y {db)^ each less than two ri^t angles, and considering the three cases in which (i) Sie ray ^ is such that all the rays that precede it belong to the first part and those which follow it to the second part, the ray ^ is followed by some rays of the first part, the ray d is preceded by some rays of the second part Application to circular arcs. If we consider an arc AB of a circle with centre Oy the points of the arc correspond univocally, and in the same order, to the rays from the point O passing through those points respectively, and the same argument by which we pa^ed from a segment of a straight line to an angle can be used to make the transition from an angle to an arc. Intersections of a straight line with a circle. It is possible to use the Postulate of Dedekind to prove that If a straight line has one point inside and one point outside a circle^ it has two points common with the circle. For this purpose it is necessary to assume (i) the proposition with reference to the perpendicular and obliques drawn from a given point to a given straight line, namely that of all straight lines drawn from a given point to a given straight line the perpendicular is the shortest, and of the rest (the obliques) that is the longer which has the longer projection upon the straight line, while those are eqiud the projections of which are equal, so that for any given length of projection there are two equal oblic^ues and two only, one on each side of the perpendicular, and (2) the proposition that any side of a triangle is less than the sum of the other two. Consider the circle (C) with centre Oy and a straight line (r) with one point A inside and one point B outside the circle. ) By the definition of the circle, if J? is /^ p AJJK B the radius, OA<Ry OB>R. Draw OP perpendicular to the straight line r. Then 0P< OAy so that OP is always less than J?, and P is therefore within the circle C Now let us fix our attention on the finite segment AB of the straight line r. It can be divided into two parts, (i) that containing all the points H for which OH<R (i.e. points inside C), and (2) that containing all the points K for which OK ^ R (points outside C or on the circumference of C). m Thus, remembering that, of two obliques from a given point to a given straight line, that is greater the projection of which is greater, we can assert that all the points of the segment PB which precede a point inside C are inside C, and those ^iYa^ follow a point on the circumference of C or outside t C are outside C. Hence, by the Postulate of Dedekind, there exists on the segment PB a f 238 ROOK I [1. Axx. point Jf such that all the points which precede it belong to the fint part and those which follow it to the second part I say that Afis common to the straight line rand the circle C, or For suppose, e.g., that OJIf< R. There will then exist a segment (or length) <r less than the difference between R and OM. Consider the point M*^ one of those which foUcw M^ such that MMT is equal to <r. Then, because any side of a triangle is less than the sum of the other two, OM' < OM^ MMT. But OM^MM'^OM-¥ir<X, whence OM'<X^ which is absurd. A similar absurdity would follow if we suppose that OM > X. Therefore OM must be equal to R. It is immediately obvious that, corresponding to the point if on the segment F£ which is common to r and C, there is another pomt on r which. has the same property, namely that which is symoaetrical to AT with respect to P. And Uie proposition is proved Intersections of two circles. We can likewise use the Postulate of Dedekind to prove that If in a givtn plane a drde C has onepoint X inside and^nepaini Y auisUe another drde C, the two cirdes intersect in two points. We must first prove the following Lemma, If Oj O' are the centres of two circles (7, C\ and J?, R' their radii respectively, the straight line 00' meets the circle C in two points A^ B^ on6 of which is inside C and the other outside it ^ Now one of these points must fall (i) on the prolongation of ffO beyond O or {2) on Off itself or (3) on the ( prolongation of Off beyond ff. (i) First, suppose A to lie on ffO pro- duced. Then Aff^AO-^OO'-^R^ Off (a). But, in the triangle Off V, ffY<OY^Off, and, since ffY>R, OY^R, R<R^Off. It follows from M that Aff>R\ and A therefore lies outside C, (2) Secondly, suppose A to lie on Off. Then Off ^OA^Aff^R^Aff ,,.{fi). From the triangle OffX we have b Off^OX^ffX, and, since OX^R^ ffX<R^ it follows that Off<R^R, whence, by {fi\ Aff <R,so that A lies inside C. I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 239 (3) Thirdly, suppose A to He on OO produced. Then R^OA^OO^OA (y). And, in the triangle OOX^ OX < 00^ OX, Y/ thatb R<.0O^^(yX, ^\ whence, by (y), oo^o'A<oa^ox, c\ or O'A < ax, and A lies inside C It is to be observed that one of the two points A, B is in the position of case (i) and the other in the position of either case (2) or case (3) : whence we must conclude that one of the two points ^, ^ is inside and the other outside the circle C\ Proof of theorem. The circle C is divided by the points A, B into two semicircles. Consider one of them, and suppose it to be described by a point moving from A to B.^ Take two separate points F, Q on it and, to fix our ideas, suppose that P precedes Q. Comparing the triangles OOP, OOQymt observe that one side OO is common, OP is equal to OQ, and the angle POO is less than the angle QOO. Therefore OP<OQ. Now, considering the semicircle APQB as divided into two parts, so that the points of the first part are inside the circle C\ and those of the second part on the circumference of C or outside it, we have the conditions necessary for the applicability of the Postulate of Dedekind (which is true for arcs of circles as for straight lines) ; whence there exists a point M separating ttu two parts, ' I say that OM^ R. \ For, if not, suppose OM < R!, ilf then o- signifies the difference between R and OM, suppose a point M\ which follows M, taken on the semicircle such that the chord MM' is not greater than o- (for a way of doing this see below). ^ Then, in the triangle OMM*, OM* < OM^ MM* < OM-¥ <r, and therefore OM' < R. It follows that AT, a point on the arc MB, is inside the circle C : which is absurd. Similarly it may be proved that OMis not greater than R. Hence OM=R. [To find a point M such that the chord MM* is not greater than <r, we may proceed thus. Draw from Mb, straight line MP distinct from OM, and cut off MP on it equal to v/i. 240 BOOK I [i. Axx. Join OP^ and draw another radius OQ such that the ang^e POQ is equal to the angle MOP. q The intersection, M\ of OQ with the circle satisfies the required condition. For MM' meets OP at right angles in 5. Therefore, in the right-ai^led triangle MSP^ MS is not greater than MP (it is less, unless MP coincides with if5, when it is equal). Therefore MS is not greater than <r/2, so that MMT is not greater than v.] J i i I I !( ft ■ B(X)K I. PROPOSITIONS. Proposition i. I On a given finite straight line to construct an equilateral triangle. I' Let AB be the given finite straight line. * Thus it is required to con- fS struct an equilateral triangle on ' the straight line A£. With centre A and distance AB let the circle BCD be described ; [Post 3] '^o again, with centre B and dis- 1 tance BA let the circle ACE I be described ; [Post 3] :. and from the point C, in which the circles cut one another, to \ the points -/4, B let the straight lines CA^ CB be joined. \ [Post. I] 15 Now, since the point A is the centre of the circle CDB^ j AC is equal to AB. [Def. 15] Again, since the point B is the centre of the circle CA£y J i BC is equal to BA. [Def. 15] But CA was also proved equal to AB ; iao therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another ; [C. N. i] therefore CA is also equal to CB. Therefore the three straight lines CA^ AB^ BC are as equal to one another. H. E. 16 BOOK I [1.1 Therefore the triangle ABC is equilateral ; and it has been constructed on the given finite straight line AB. (Being) what it was required to do. I. On a given finite straight line. The Greek usage diffen finom oars in that the definite article is employed in sudi a phrase as this where we have the indefinite. M rft do$€lfffit c^ffittf w€W€pafffUpiit, **on iJkg given finite straight line," i.e. the finite straight line which we choose to take. 3. Let AB be the given finite straight line. To be strictly literal we should have to translate in the reverse order "let the given finite straight line be the (straight line) AB^i bat this order is inconvenient in other cases where there is more than one datam, e^ in the sttting-aut of I. 3, *'let the given point be A^ and the given straight line BC^** tihe awkward- ness arising from the omission of the verb in the second daose. Hence I have, for deamess' sake, adopted the other order throagfaoat the book. 8. let the circle BCD be described. Two things are here to be noted, (1) the elegant and practically universal use of the perfect passive imperative in constructions, yrfp £ f $t » meaning of course "let it Aatfe heen described " or "suppose it described,** (3) the impossi- bility o7 expressing shortly in a translation the force of the words in their original order. jci^irXot ytypi^ta 6 BFA means literally "let a circle have been described, the (circle, namehr, which I denote bv) BCD" Similarly we have lower down *' let straight lines, (namely) the (straight lines) CA, CB, be joined,'* iwttfB^Stfaap MtUu td FA, r£ There aecms to be no practicable alternative, in En^i^ bat to translate as I have done in the teat. 13. from the point C... Eodid is carefid to adhere to the phraseoloMpr of Pottalate i except that he speaks of "joining" (^rc^r^dMvar) instead of " drawing '^(W^^tiv). He does not allow himself to use the shortenei expression " let the straight line r^ be joined " (without mention of the points /*, C) until i. 5. 10. each of the straight lines CA, CB, inripa rflp TA, FB and 94. the three straight lines CA, AB, BC, ol rpeit al FA, AB, BF. I have, here and in all similar expressions, inserted the words "straight lines" which are not in the Greek. The possettion of the inflected definite article enables the Greek to omit the words, but tlds is not possible in English, and it would scarcely be Ei^lish to write "each of CA, CB" or "the three CA, AB, Bc:' It is a commonplace that Euclid has no right to assume, without pre- mising some postulate, that the two circles wtJl meet in a point C To supply what is wanted we must invoke the Principle of Continuity (see note thereon above, p. 235). It is sufficient for the purpose of this proposition and of I. 22, where there is a similar tacit assumption, to use the form of postulate suggested by Killing, ''/if ^ ^ne [in this case e.g^ the circumference AC£] belongs entirely to a figure [in this case a plane] which is divided into two parts [namely the part enclosed within the circumference of the circle BCD and the part outside that circle], and if the line has at least one point common with each part, it must also meet the boundary between the parts [Le. the circum- ference ACE must meet the circumference BCLf\,^^ Zeno's remark that the problem is not solved unless it is taken for granted that two straight lines cannot have a common s^ment has already been mentioned (note on Post 2, p. 196). Thus, if AC, BC meet at F before reaching C, and have the part FC common, the triangle obtained, namely FAB, will not be equilateral, but FA, FB will each be less than AB, But Post 2 has already laid it down that two straight lines cannot have a common segment Proclus devotes considerable space to this part of Zeno's criticism, but satisfies himself with the bare mention of the other part, to the effect that it is also necessary to assume that two circumferences (with different centres) cannot have a common part That is, for anything we know, there may be any number of points C common to the two circumferences ACE, BCD. It is not until in. 10 that it is proved that two circles cannot intersect in more < il 1. 1] PROPOSITION I 243 points than two, so that we are not entitled to assume it here. The most we can say is that it is enough for the purpose of this proposition if one equilateral triangle can be found with the given base ; that the construction only gives two such triangles has to be left over to be proved subsequently. And indeed we have not long to wait ; for i. 7 clearly shows that on either side of the base AB only one equilateral triangle can be described. Thus i. 7 gives us the numlfer of solutions of which t^e present problem is susceptible, and it supplies the same want in i. 22 where a triangle has to be described with three sides of given length ; that is, i. 7 furnishes us, in both cases, with one of the essential parts of a complete Siapur|A<^ which includes not only the determination of the conditions of possibility but also the number of solutions (voo-axolf ^x^P<^ Proclus, p. 202, 5). This view of i. 7 as supplying an equivalent for iii. 10 absolutely needed in i. i and i. 22 should serve to correct the idea so common among writers of text-books that i. 7 is merely of use as a lemma to Euclid's proof of i. 8, and therefore may be left out if an alternative proof of that proposition is adopted. Agreeably to his notion that it is from i. i that we must satisfy ourselves that isosceles and scalene triangles actually exist, as well as equilateral triangles, Proclus shows how to draw, first a particular isosceles triangle, and then a scalene triangle, by means of the figure of the proposition. To make an isosceles triangle he produces AB in both directions to meet the respective circles in Z?, E^ and then describes circles with A^ B ais centres and AE^ BD as radii respectively. The result is an isosceles triangle with each of two sides double of the third side. To make an isosceles triangle in which the equal sides are not so related to the third side but have any given length would require . the use of i. 3 ; and there is no object in treating the question at all in advance of I. 22. An easier way of satisfying our- selves of the existence of some isosceles triangles would surely be to conceive any two radii of a circle drawn and their extremities joined. There is more point in Proclus* construction of a scalene triangle. Suppose ^C to be a radius of one of the two circles, and D a point on AC lying in that portion of the circle with centre A which is outside the circle with centre B, Then, joining BD^ as in the figure, we have a triangle which obviously has all its sides unequal, that is, a scalene triangle. The above two constructions appear in an-Nairlzfs commentary under the name of Heron; Proclus does not mention his source. In addition to the above construction for a scalene triangle (producing a triangle in which the ''given" side is greater than one and less than the other of the two remaining sides), Heron has two others showing the other two possible cases, in whic^ the '' given " side is (i ) less than, (2) greater than, either of the other two sides. 16 — 2 UA BOOR I Proposition 2. [Postal To place at a given paint (as an extremity) a straight Une equal to a given straight line. Let A be the given point, and BC the given straight line. Thus it is required to place at the point A (as an extremity) 5 a straight line equal to the given straight line BC. From the point A to the point B let the straight line AB be joined ; [Post i] and on it let the equilateral triangle JO DAB be constructed. [i. i] Let the straight lines AE, BF be produced in a straight line with DA^ DB\ [Post a] with centre B and distance BC let the 15 circle CGH be described ; and again, with centre D and distance DG let the circle GKL be described. [Post 3] Then, since the point B is the centre of the circle CGH^ BC is equal to BG. so Again, since the point D is the centre of the circle GKL, DL is equal to DG. And in these DA is equal to DB ; therefore the remainder AL\s equal to the remainder BG. \C.N. 3] But BC was also proved equal to BG ; therefore each of the straight lines AL, BC is equal \oBG. And things which are equal to the same thing are also equal to one another ; [C.AI i] J therefore AL is also equal to BC. Therefore at the given point A the straight line AL is placed equal to the given straight line BC. (Being) what it was required to do. I. (as an extremity). I have inserted these words because **to place a stiaisht line at a given point '* (vp^t r^ Ba$4vri ^Mc(^)is not quite clear enough, at least in English. la Let the straight lines AE, BP be produced.... It will be observed ^t in this first application of Postulate 3, and a^ ain in i. 5, Euclid speaks of the amHnuaiiom ci the straight Une as that which is produced in such cases, iKfi€p\^waaif and wpoatxPtfiK^^Sttntw meaninglittle more than drawing straight lines " in a straight line with " the given straight lines. The first place in which Euclid uses phraseology exactly corresponding to onn when as I r 1.2] PROPOSITION 2 24s speakii^ of a straight line being produced is in i. 16 :" let one tide of it, BC^ be produced to Z> " {wpofftKPtpX^Bw a^oO fda, wXevpii iiBT iwlrb A). 22. the remainder AL...the remainder BO. The Greek expressions are Xocr^ ^ j AA and Xocrj rv BH, and the literal translation would be '*AL (or BG) remaining,** ! but the shade of meaning conveyed by the position of the definite article can hardly be expressed in English. This proposition gives Proclus an opportunity, such as the Greek \ commentators revelled in, of distinguishing a multitude of ^ases. After I explaining that those theorems and problems are said to have cases which • have the same force, though admitting of a number of different figures, and t preserve the same method of demonstration while admitting variations of ^ position, and that cases reveal themselves in the construction^ he proceeds to distinguish the cases in this problem arising from the different positions which the given point may occupy relatively to the given straight line. It may be (he says) either (i) outside the line or (2) on the line, and, if {i\ it may be \ either {a) on the line produced or {b) situated obliquely with regard to it ; if . (2), it may be either {a) one of the extremities of the line or {b) an intermediate > point on it It will be seen that Proclus* anxiety to subdivide leads him to I give a "case," (2) (a), which is useless, since in that "case" we are given what we are re(}uired to find, and there is really no problem to solve. As I Savile says, " qui quaerit ad p punctum ponere rectam aequalem r^ fiy rectae, j quaerit quod datum est, quod nemo faceret nisi forte insaniat" ! Proclus gives the construction for (2) (b) following Euclid's way of taking t (? as the point in which the circle with centre B intersects DB produced^ and then proceeds to " cases," of which there are still more, which result from the different ways of drawing the equilateral triangle and of producing its sides. • This last class of " cases " he subdivides into three according as AB is (i) equal to, (2) greater than or (3) less than BC, Here again " case " (i) serves no purpose, since, if AB is equal to BC^ the problem is already solved. But Proclus' figures for the other two cases are worth giving, because in one of them the point G is on BD produced beyond Z>, and in the other it lies on BD itself and there is no need to produce any side of the equilateral triangle. A glance at these figures will show that, if they were used in the proposition, each of t}iem would require a slight modification in the wording (i) of the construction, since BD is in one case produced beyond D instead of B and in the other case not produced at all, (2) of the proo^ since BG, instead of being the difference between DG and DBy is in one case the sum of DG and DB and in the other the difference between DB and DG. 346 BOOK I Modem editors generaUy seem to dassify the cues aooofding to Ae possible variations in the construction rather than according to differences in the data. Thus Lardner, Potts, and Todhunter distinguiui eight cases due to the three possible alternatives, (i) that the given point may be joined to either end of the given straight line, (2) that the equilateral triangle may then be described on either side of the joining line, and (3) that the side of the equilateral triangle which is produced may be produced in either direction. (But it should Imve been observed that, where AB is greater than BC^ tfie third alternative is between producing DB and not producing it at all) Potts adds that, when the given pomt lies either on the line or on the line pioduoed, the distinction which arises from joining the two ends of the line with the given point no longer exists, and there are only four cases of the problem (I think he should rather have said sohiHons). To distinguish a number of cases in this way was foreign to the really classical manner. Thus, as we shall see, Euclid's method is to give one case only, for choice the most difficult, leaving the reader to suf^ly the rest for himself. Where there was a real distinction between cases, sufficient to necessitate a substantial difference in the proof, the practice was to gjve separate enunciations and -proofs altogether, as we may see, e.g., from the Conies and the De sectione ratioms of Apollonius. Proclus alludes, in conclusion, to the error of those who proposed to solve I. 2 by describing a circle with the given point as centre and with a distance equcU to J9C, which, as he says, is a ptHtio prindpH. De Morgan puts the matter very clearly {Supplementary Remarks on the first six Books of EudS^s Elements in the Companion to the Almanac^ 1349, P- 6). We should ^'insist," he says, ''here upon the restrictions imposed by the first three postulat^ which do not allow a circle to be drawn with a compass-carried distance; suppose the compasses to close of themselves the moment they cease to toudi the paper. These two propositions [i. 2, 3] extend the power of construction to what it would have been if all the usual power of the compasses had been assumed ; they are mysterious to all who do not see that postulate iii does not ask for every use of the compasses P Proposition 3. Given two unequal straight lines^ to cut off front the greater a straight line equal to the less. 5. Let ABy C be the two given un- equal straight lines, and let AB be the greater of them. Thus it is required to cut off from AB the greater a straight line equal to C the less. At the point A let AD be placed equal to the straight line C ; [i. 2] and with centre A and distance AD let the circle DEF be described. [Post 3] *\ ! 1.3,4] PROPOSITIONS 2—4 247 Now, since the point A is the centre of the circle DEF, AE is equal to AD. [Def. 15] But C is also equal to AD. Therefore each of the straight lines AE, C is equal to AD ; so that AE\s also equal to C. \C.N, i] Therefore, given the two straight lines AB, C, from AB the greater A[E has been cut off equal to C the less. (Being) what it was required to do. Proclus contrives to make a number of ''cases" out of this proposition also, and gives as many as eight figures. But he only produces this variety by practically incorporating the construction of the preceding proposition, instead of assuming it as we are entitled to do. If Prop. 2 is assumed, there is really only one " case " of the present proposition, for Potts' distinction between two cases according to the particular extremity of the straight line from which the given length has to be cut off scarcely seems to be worth making. Proposition 4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles 5 will be equal to the remaining angles respectively, namely those which the equal sides subtend Let ABC, DEE he two triangles having the two sides AB, AC equal to the two sides DE, Z?-F respectively, namely AB to DE and AC to DE, and the angle BAC equal to the ID angle EDE. I say that the base BC is also equal to the base EE, the triangle ABC will be equal to the triangle DEE, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that 15 is, the angle ABC to the angle DEE, and the angle ACB to the angle DEE. For, if the triangle ABC be applied to the triangle DEE, and if the point A be placed 20 on the point D and the straight line AB on DE, then the point B will also coincide with E, because AB is equal to DE. 348 BOOK I [1.4 5 Again, AB coinciding with DE^ the straight line AC will also coincide with DF, because the angle BAC is equal to the angle EDF\ hence the point C will also coincide with the point F^ because AC\s again equal to DF. o But B also coincided with E ; hence the base BC will coincide with the base EF. [For if, when B coincides with E and C with F, the base BC does not coincide with the base EF, two straight lines will enclose a space : which is impossible. 5 Therefore the base BC will coincide with EF^ and will be equal to it \C.N. 4] Thus the whole triangle ABC will coincide with the whole triangle DEF, and will be equal to it. p And the remaining angles will also coincide with the remaining angles and will be equal to them, the angle ABC to the angle DEF, and the angle ACB to the angle DFE. Therefore etc. ^5 (Being) what it was required to prove I — 3. It is a fiict that Eadid*s enunciations not infreauently leave something to be desued in point of clearness and precision. Here he speaks of the triangles having "the angle equal to the angle, namely the angle contained by the equal straight lines ** (rV Ttfrfor ri yttwlg, Uniff fxtl '''¥ ^ f^ ^^^ t^tiQ^ vcptcxo/i^nyr), only one of the two angles being described in the latter expression ^n the accusative), and a similar expression in the dative being left to be understood of the other angle. It is curious too that, after mentioning two "sides" he speaks of the angles contained by the equal ** straight lines," not ^'sidesy It majr be that ne wished to i^ere scrupulously, at the outset, to the phraseolof^ of the dennitions, where the angle is the inclination to one another of two iines or strait lines. Similarljr in the enunciation of i. 5 he speaks of producing the equal '* straight lines " as if to keep stnctly to the wording of Postulate 3. 3. respectively. I agree with Mr H. M. Taylor {Euclid, p. ix) that it is best to abandon the traditional translation of "each to each,' which would naturally seem to imply that all the four magnitudes are equal rather than (as the Greek iKaHpa ixaiip^ does) tnat one is eoual to one and the other to the other. 3. tne base. Here we have the word ^au used for the first time in the Elements, Proclus explains it (p. 336, 13 — 15) as meaning (i), when no side of a triangle has been mentioned oefore, the side ** which is on a level with the sight *' (rV rp^ rg 6^ K»fi4nfw), and (3), when two sides have already been mentioned, the third side. Ftodus thus avoids the mistake made by some modem editors who explain the term exclusively with reference to the case where two sides have been mentioned before. That this is an error is proved (i) by the occurrence of the term in the enunciations of i. 37 etc. about triangles on tne same base and equal bases, (3) by the application of the same term to the bases of parallelograms in I. 35 etc. The truth is that the use of the term must have been suggested oy the practice of drawing the particular side horizontally, as it were, and the rest of the figure above it. The dose of a figure was therefore spoken of, primarily, in the same sense as ue base of anything ■( J ! 1.4] PROPOSITION 4 249 else, e.g. of a pedestal or column; but when, as in i. 5, two triangles were compared occupying other than the normal positions which gave rise to the name, and when two sides had been previously mentioned, the base was, as Proclus says, necessarily the third side. 6. subtend, itwvrtlwtup ifw6, **to stretch under," with accusative. 9. the angle B AC. The fiill Greek expression would be ^ ^6 rwv BA, AF v€pux90uhii YMrto, "the angle contained by the (straight lines) BjI^ AC." But it was a common practice of Greek geometers, e.g. of Archimedes and Apollonius (though not apparently of Euclid), to use the abbreviation «l BAP for «l BA, AF, **the (straight lines) BA, AC.** Thus, on «-cptcxoA</n| being dropped, the expression would become first ^ M rtDv BAT Tw^fo, then 4 ^6 BAT ywla, and finally ^ inr6 BAP, without ytatda, as we rqrolarly find it in Euclid. 17. if the triangle be applied to..., 33. coincide. The difference between the technical use of the passive i^apfU^taBcu "to be a^piitd {to)" and of the active i^apfUftuf "to eeindde (with) '^ has been noticed above (note on Common Notion 4, pp. 334 — 5). J 3. [For \i, when B coincides... ^6. coincide with EF]. Heiberg {P^troHpcminmau fid in Hermis, xxxviii., 1903, p. 50) has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-Nairld {Codex Leidensis ^99, 1, ed. Besthom-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.B.D., which shows that they constitute a scholium only. They were doubtless added bjjr some commentator who thought it necessary to explain the immediate inference that, smce B coincides with E and C with F, the straight line BC coincides with the straight line RF, an inference which really follows from the definition of a straight line and Post. 1 ; and no doubt the Postulate that " Two straight lines cannot enclose a space " (afterwards placed among the Common Notions) was interpolated at the same time. 44. Therefore etc. Where (as here) Euclid's conclusion merely repeats the enunciation ( word for word, I shall avoid the repetition and write " Therefore etc.*' simply. I In the note on Common Notion 4 I have already mentioned that Eudid ^ obviously used the method of superposition with reluctance, and I have given, after Veronese for the most part, the reason for holding that that method is not admissible as a theoretical means of proving equality, although it may be of use as 9^ practical test, and may thus furnish an empirical basis on which to found a postulate. Mr Bertrand Russell observes {Principles of Maihemaiics I. p. 405) that Euclid would have done better to assume i. 4 as an axiom, as . is practically done by Hilbert {Grundlagen der Geometric^ p. 9). It may be ( / that Euclid himself was as well aware of the objections to the method as are i ' his modem critics ; but at all events those objections were stated, with almost \^ equal clearness, as early as the middle of the i6th century. Peletarius (Jacques Peletier) has a long note on this proposition (In EucHdis EUmemta j geometrica demonstrationum libri sex^ i557)> in which he observes that, if I' superposition of lines and figures could be assumed as a method of proof^ the \ whole of geometry would be full of such proofs, that it could equally well have 4 been used in i. 2, 3 (thus in i. 2 we could simply have supposed the line taken s up zxA placed at the point), and that in short it is obvious how far removed the ' method is from the dignity of geometry. The. theorem, he adds, is obvious in itself and does not require proof ; although it is introduced as a theorem, it would seem that Euclid intended it rather as a definition than a theorem, '^for I cannot think that two angles are egual unless I have a conception of what equality of angles is.'' Why then did Euclid include the proposition among theorems, instead of placing it among the axioms ? Peletanus makes the best excuse he can, but concludes thus: '' Huius itaque propositionis veritatem non aliunde quam a communi iudido petemus; cogitabimusque figuras figuris superponere, Mechanicum quippiam esse: intelUgere verb, id demum esse Mathemadcum.'' Expressed in terms of the modem systems of Congruence-Axioms referred to in the note on Common Notion 4, what Euclid really assumes amounts to »' the following : (i) On the line DE^ there is a point E^ on either side of 27, such that AB is equal to DE. i 250 BOOK I [1.4 (2) On either side of the ray DE there is a ray DF such that the angle EDF'v^ equal to the angle BAC. It now follows that on DF there is a pcnnt F such that DF is equal to^C. And lastly (3), we require an axiom from which to infer that the two remaining angles of the triangles are respectively equal and that the bases aie equal. I have shown above (pp. 229—330) that Hilbert has an axiom stating the equality of the remaining angles simply, but proves the equality of the b^^ses. Another alternative is that of Pasdi ( Variesungm uber neuert Geomdrk^ p. 109) who has the following ''Grundsatz": If two figures AB and FGH are given (FGH not being contained in a straight length), and AB^ FG are congruent and if a plane surfoce be laid through A and B^ we can specify in this plane surface, produced if necessary, two points C, Z>, neither more nor less, such that the figures ABC and ABD are congruent with the figure FGH^ and the straight Tine CD has with the straight line AB or with AB produced one point common. I pass to two points of detail in Euclid's proof: (i) The inference that, since B coincides with E^ and C with F^ the bases of the triangles are wholly coincident rests, as expressly stated, on the impossibility of two straight lines enclosing a space, and therefore presents no difficulty. But (2) most editors seem to have fiuled to observe that at the very beginning of the proof a much more serious assumption is made without any explanation whatever, namely that, if ^^ be placed on D^ and AB on DE^ the pomt B will coincide with ^, because AB is equal to DE. That is, the converse of Common Notion 4 is assumed for straight lines. Produs merely observes, with regard to the converse of this Common Notion, that it is only true in the case of things "of the same form " (6/ioc(S^), which he explains as meaning stnught lines, arcs of one and the same circle, and angles '' contained by lines simiku: and similarly situated" (p. 241, 3 — 8^. Savile however saw the difficulty and grappled with it in his note on the Common Notion. After stating that all straight lines with two points common are congruent between them (for otherwise two straight lines would enclose a space), he argues thus. Let there be two straight lines AB^ DE^ and let A be placed on 2?, and AB on DE. Then B will coincide with E. For, if not, let B fall somewhere short of E or beyond E ; and in either case it will follow that the less is equal to the greater, which is impossible. Savile seems to assume (and so apparently does Lardner who gives the same proof) that, if the straight lines be "applied," B will fall somewhere on DE or DE produced. But the ground for this assumption should surely be stated ; and it seems to me that it is necessary to use, not Postulate i alone, nor Postulate 2 alone, but both, for this purpose (in other words to assume, not only that two straight lines cannot enclose a space^ but also that two straight lines cannot have a common segment). For the only safe course is to place A upon D and then turn AB about D until some point on AB intermeduite between A and B coincides with some point on DE. In this position AB an^ DE have two points common. Then Postulate i enables us to infer that the straight lines coincide between the two common points, and Postulate 2 that they coincide beyond the second common point towards B and E. Thus the straight lines coincide throughout so fieur as both extend; and Savile's argument then proves that B coincides with E. J i.,5] PROPOSITIONS 4, 5 2^1 Proposition 5. In isosceles triangles the angles at the base are equal to one another^ and, if the equal straight lines be produced further, the angles under the base will be equal to one another. Let ABC be an isosceles triangle having the side AB 5 equal to the side AC\ and let the straight lines BD, CE be produced further in a straight line with AB, AC. [Post 2] 1 say that the angle ABC is equal to the angle ACB^ and the angle CBD to the angle BCE. 10 Let a point F be taken at random on BD] from AE the greater let AG he cut off equal to AF the less ; [i. 3] and let the straight lines FC, GB be joined. [Post i] 15 Then, since AF is equal to -^G^ and ^^to^C, the two sides FA^ AC are equal to the two sides GA, A By respectively ; and they contain a common angle, the angle FAG. 20 Therefore the base FC is equal to the base GB, and the triangle AFC is equal to the triangle AGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ACF to the angle ABG, 25 and the angle AFC to the angle AGB. [i. 4] And, since the whole AFis equal to the whole AG, and in these AB is equal to AC, the remainder BF is equal to the remainder CG. But FC was also proved equal to GB ; 30 therefore the two sides BF, FC are equal to the two sides CG, GB respectively ; and the angle BFC is equal to the angle CGB, while the base BC is common to them ; therefore the triangle BFC is also equal to the triangle CGB, 35 and the remaining angles will be equal to the remaining [ 353 BOOK I [l 5 angles respectively, namely those which the equal sides subtend ; therefore the angle FBC is equal to the angle GCB^ and the angle BCF to the angle CBG. ¥> Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG is equal to the angle BCF, the remaining angle ABC is equal to the remaining angle ACB; 15 and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB ; and they are under the base. Therefore etc. Q. E. d. 1. the equal straight lines (meaniiig the equal sides). Cf. note on the amilar expression in Frop. 4, lines 1, $. la Let a point P be taken at random on BD, c2X4^^ M rft BA rvx^ tfit^wSbr H E, where rvx^v aii/uw means "a chance point.** 17. the two sides FA, AC are equal to the two sides QA, AB respccthreljr, Mo ol Zl, AT ival nut HA, AB Krai tlaiw ixmripa inrip^. Here, uid in nnmberlest later passages, I have inserted the word "ades** for the reason given in the note on I. i, fine so. It would have been permissible to sopply either '* straight lines" or "ades**; but on the whole "sides '* seems to be more in accordance with the phrasedogjr of I. 4. 33. the base BC is common to them, i.e., apparently, common to the atngUSf as the odrc^v in fidatt ai^rwr icoirii can only refer to ytpUt and yvAa preceding. Simson wrote **and the base ^C is common to the two triangles BFC^ CGB*'\ Todhunter left out these words as being of no use and tending to perplex a beginner. But Euclid evidently chose to quote the conclusion of I. 4 exactly ; the first phrase of that conclusion is that the bases (of the two triangles) are equal, and, as the equal bases are here the sams base, Euclid naturally substitutes the word "common** for "equal." 48. As ** (deing) what it was required to prove '* (or " do '*) is somewhat long, I shall henceforth write the time-honoured "Q. E. D.' and "Q. E. F." for Swtp idu 9€t^ and 9rtp According to Proclus (p. 250, 20) the discoverer of the fact that in any isosceles triangle the angles at the base are equal was Thales, who however is said to have spoken of the angles as being similar^ and not as being equal, (Cf. Arist De caelo iv. 4, 31 1 b 34 irpos hyjom -^^wm ^v€rai ^cpoficyoy where egua/ angles are meant.) A pre-Euclidean proof of I. 5. One of the most interesting of the passages in Aristotle indicating differences between Euclid's proofs and those with which Aristotle was fieunilmr, in other words, those of the text-books immediately preceding Euclid's, has reference to the theorem of i. 5. The pissage (Anal. Prior, i. 24, 41 b 13—22) is so important that I must quote it in full. Aristotle is illustrating the fact that in any syllogism one of the propositions must be affirmative and universal (ffoMXov). ''This," be says, ''is better shown in the case of geometrical propositions ^ i^ roTs &aypafi/Aao-tv), e.g. the proposition that the angles at the base of an isosceles triangle are equal, "For let ^, ^ be drawn [i.e. joined] to the centre. I. S] PROPOSITION 5 «53 ''If, then, we assumed (i) that the angle AC [Le. ^ + C] is equal to die angle BD [Le. B-^D^ without asserting generally that the angles of semidrcUs are equals and again (2) that the angle C is equal to the angle D without making the further assumption that the two angles of ali segments are equals and if we then inferred, lastly, that, since the whole angles are equal, and equal angles are subtracted from them, the angles which remain, namely ^, ^ are equal, we should commit a petitio prinapii'^ Some peculiarities of phraseology will be observed in this passage. (i) A^ B are said to be drawn (4yfiiyai) to the centre (of the cirde of which the two equal sides are radii) asif A^B were not the angular points but the sides or the radii themselves. There is at least one pandlel for this in Euclid (cf. IV. 4). (2) "The angle AC*' is the angle which is the sum of A and C, and A means here the angle at A of the isosceles triangle shown in the figure, and afterwards spoken of by Aristotle as £, while C is the " mixed " angle between AB and the circumference of the smaller segment cut off by it (3) The ''angle of a semicircle" (i.e. the "angle" between the diameter a^d the circumference, at the extremity of the diameter) and the "angle ^ a segment" appear in Euclid in. 16 and in. Def. 7 respectively, obviously as survivals from earlier text-books. ■ .^.-^ But the most significant facts to be gathered from the extract are that in the text-books whidb preceded Euclid's " mixed " angles played a much more important part than they do with Euclid, and, in particular, that at least two "^ propositions concerning such angles appeared quite at the beginning, namdy "x the propositions that the (mixed) angles of semicircles are equal and that the two .% (fkixed) angles of any segment of a circle are equal. The wording of the first . ;: ; of the two propositions is vague, but it does not necessarily mean more dian ' V-^ that the two (mixed) angles in one semicircle are equal, and I know of no -J; evidence going to show that it asserts that the angle of any one semicircle is ^^ equal to the angle of any other semicircle (of different size). It is quoted in _ "^ the same form, " because the angles of semicircles are equal," in die Latin n^ translation from the Arabic of Heron's Catoptrica^ Prop. 9 (Heron, VoL VL^ '^ Teubner, p. 334), but it is only inferred that the different radii of one drde *^ V? make equal "angles" with the circumference ; and in the similar proposition --^) of the Pseudo-Euclidean Catcptrica (Euclid, Vol. vii., p. 294) ai4;les of the > same sort in one circle are said to be equal "because they are (angles) of "'^ a semicircle." Therefore the first of the two propositions may be cmly a ' \ particular case of the second. . t> But it is remarkable enough that the second proposition (that the two "^ " CMgles of any segment of a circle are equal) should, in earlier text-books, have ;^ : been placed before the theorem of Eucl. i. 5. We can hardly suppose it to ^> have been proved otherwise than by the superposition of the semicndes into which the circle is divided by the diameter which bisects at right angles the _^ A base of the segment; and no doubt the proof would be closely connected 1^ "* that of Thales' other proposition that any diameter of a circle bisects it, which must also (as Proclus indicates) have been proved by superposing one of the two parts upon the other. It is a natural inference fix>m the passage of Aristode that Euclid's proof of aS4 BOOK I D- 5 I. 5 was his own, and it would thus appear that his innovations as regards order of propositions and methods of proof began at the very threshold of the subject Proof without producing the sides. In this proof, given by Proclus (pp. 248, a a — 349, 19), D and E are taken on AB^ AC, instead of on AB, Atprodttad, so that AD, AEwn, equal The method of proof is of course exactly like Euclid's, but it does not establish the equality of the angles beyond the base as well. Pappus' proof. Proclus (pp. 249, 20—250, 12) says that Pappus proved the theorem in a still shorter manner without the help of any construction whatever. This very interesting proof is given as follows : " Let ABC be an isosceles triangle, and AB equal to AC. Let us conceive this one triangle as two triangles, and let us arfl;ue in this way. Smce AB is equal to AC, and AC to AB, the two sides AB, AC are equal to the two sides AC, AB. And the angle BAC is equal to the angle CAB, for it is the same. Therefore all the corresponding parts (in the triangles) are equal, namely BC to BC, the triangle ABC to the triangle ABC (Le. ACB), the angle ABC to the angle ACB, and the angle ACB to the angle ABC, (for these are the angles subtended by the equal sides AB, AC. Therefore in isosceles triangles the angles at the base are equal." This will no doubt be recognised as the foundation of the alternative proof frequently given by modem editors, though they do not refer to Pappus. But they state the proof in a different form, the common method being to suppose the triangle to be taken up, turned over, and placed again upon stsetf, after which the same considerations of congruence as those used by Euclid in I. 4 are used over again. There is the obvious difficulty that it supposes the triangle to be taken up and at the same time to remain where it is. (Cf. Dodgson's humorous remark upon this, Euclid and his modem Rivals, p. 47.) Whatever we may say in justification of the proceeding (e.g. that the triangle may be supposed to leave a trace), it is really equi^ent to assuming &e construction (hypothetical, if you will) of another triangle equal in all respects to the given triangle ; and such an assumption is not in accordance with Euclid's principles and practice. It seems to me that the form given to the proof by Pappus himself is by &r the best, for the reasons (i) that it assumes no construction of a second triangle, real or hypothetiad, (3) that it avoids the distinct awkwardness involved by a proof which, instead of merely quoting and applying the result of a previous proposition, repeats, with reference to a new set of data, the process by which that result was established. If it is asked how we are to realise Pappus' idea of two triangles, surely we may answer that we keep to one triangle and merely view it in two aspects. If it were a question of helping a b^;inner to understand this, we might say that one triangle is the triangle \ t J I I. s, 6] PROPOSITIONS 5, 6 255 looked at in front and that the other triangle is the same triangle looked at from behind] but even this is not really necessary. Pappus' proof, of course, does not include the proof of the second part of the proposition about the angles under the base, and we should still have to establish this much in the same way as Euclid does. Purpose of the second part of the theorem. An interesting question arises as to the reason for Euclid's insertion of the second part, to which, it will be observed, the converse proposition i. 6 has nothing corresponding. As a matter of fact, it is not necessary for any subsequent demonstration that is to be found in the original text of Euclid, but only for the interpolated second case of i. 7; and it was perhaps not unnatural that the undoubted genuineness of the second part of i. 5 convinced many editors that the second case of i. 7 must necessarily be Euclid's also. Proclus' explanation, which must apparently be the right one, is that the second part of i. 5 was inserted for the purpose of fore-arming the learner against a possible objection (Ivoroo-ic), as it was technically called, which might be raised to i. 7 as given in the text, with one case only. The objection would, as we have seen, take the specific ground that, as demonstrated, the theorem was not conclusive, since it did not cover all possible cases. From this point of view, the second part of l 5 is useful not only for i. 7 but, according to Proclus, for i. 9 also. Simson does not seem to have grasped Proclus' meaning, for he says : " And Proclus acknowledges, that the second part of Prop. 5 was added upon account of Prop. 7 but gives a ridiculous reason for it, 'that it might afford an answer to objections made against the 7th,' as if the case of the 7th which is left out were, as he expressly makes it, an objection against the proposition itself." Proposition 6. If in a triangle two angles be equal to one another^ the sides which subtend the equal angles will also be equal to one another. Let ABC be a triangle having the angle ABC equal to the angle ACB\ I say that the side AB is also equal to the ^ side AC. j For, if AB is unequal to AC, one of them is greater. Let AB be greater; and from AB the greater let DB be cut off equal to -^C the less ; let DC be joined. Then, since DB is equal to AC, and BC is common, the two sides DB, BC are equal to the two sides AC, CB respectively ; 956 BOOK I [l6 and the angle DBC is equal to the angle ACB ; therefore the base DC is equal to the base AB^ and the triangle DBC will be equal to the triangle ACB^ the less to the greater : which is absurd Therefore AB is not unequal to AC; it is therefore equal to it Therefore etc. Q. E. D. Euclid assumes that, because Z> is between A and B, the triangle DBC 13 less than the triangle ABC. Some postulate is necessary to justify diis tadt assumption; considering an angle less than two right angles, say the angle ACB in the figure of tiie pr(qx)sition, as a cluster df rays issuing fiom C and bounded by the rays CA^ CB^ and joining AB (where A,B$xe any two points on CA^ CB respectively), we see that to each successive ray taken in the direction from CA to CB there corresponds one point on AB in whidi the said ray intersects AB^ and that all the points on AB taken in <mler from A to B correspond univocally to all the ra^ taken in order frcun CA to CB, each point namely to the ray intersectmg AB in the point We have here used, for the &rst time in the ElementSf the method of reductio ad absurdum^ as to which I would refer to the section above (pp. 136, 140) dealing with this among other technical terms. This proposition also, being the converse of the preceding proposition, brings us to the subject of Geometrical Conversion. This must of course be distinguished from the logical conversion of a proposition. Thus, from the proposition that all isosceles triangles have the angles opposite to the equal sides ecjual, logical conversion would only enable us to conclude that some triangles with two angles equal are isosceles. Thus I. 6 is the geometrical, but not the logical, converse of i. 5. On the other hand, as De Morgan points out {Companion to the Almanac, 1849, p. 7), L 6 is a purely ^Ss^o/ deduction from i. 5 and i. 18 taken together, as is i. 19 aba For the general argument see the note on i. 19. For die present proposition it is enough to state the matter thus. Let X denote the class of triangles which have the two sides other than the base equal, Y the class of triangles which have the base angles equal \ then we may call non-A' the class of triangles having the sides other than the base unequal, non- Y the class of triangles having the base angles unequal Thus we have All AT is K, [i. s] All lion-X is non-K; [i. 18] and it is a purely logical deduction that All y is AT. [i. 6] According to Produs (p. 352, 5 sqq.) two forms of geometrical conversion were distinguished. (i) The leading form {jrpmfyoviUrq), the oonvemoTi par excellence (1I1 KVfimi 1 f 1.6] PROPOSITION 6 15? ayruFTfio^, is the complete or simple conversion in which the hypothesis and the conclusion of a theorem change places exactly, the conclusion of the theorem bein^ the hypothesis of the converse theorem, which again establishes, as its conclusion, the hypothesis of the original theorem. The relation between the first part of i. 5 and i. 6 is of this character. In the former the hypothesis is that two sides of a triangle are equal and the conclusion is that the angles at the base are equal, while the converse (i. 6) starts from the hypothesis that two angles are equal and proves that the sides subtending them are equal. (2) The other form of conversion, which we may call partial^ is seen in cases where a theorem starts from two or more hypotheses combined into one enunciation and leads to a certain conclusion, liter which the converse theorem takes this conclusion in substitution for one of the hypotheses of the original theorem and from the said conclusion along with the rest of the original hypotheses obtains, as its conclusion, the omitted hypothesis of the original theorem, i. 8 is in this sense a converse proposition to i. 4 ; for i. 4 takes as hypotheses (i) that two sides in two triangles are respectively equal, (3) that the included angles are equal, and proves (3) that the bases are equal, while I. 8 takes (i) and (3) as hypotheses and proves (2) as its conclusion. It is clear that a conversion of the leading type must be unique, while there may be many partial conversions of a theorem according to the number of hjrpotheses from which it starts. Further, of convertible theorems, those which took as their hypothesis the gtnus and proved a property were distinguished as the leading theorems (irpoi;yov/tcva), while those which started from the property as hypothesis and described, as the conclusion, the genus possessing that property were the converse theorems, i. 5 is thus the leading theorem and i. 6 its converse, since the genus is in this case taken to be the isosceles triangle. Converse of second part of I. 5. Why, asks Proclus, did not Euclid convert the second part of i. 5 as well ? He suggests, properly enough, two reasons: (i) that the second part of i. 5 itself is not wanted for any proof occurring in the original text, but is only put in to enable objections to the existing form of later propositions to be met, whereas the converse is not even wanted for this purpose \ (2) that the converse could be deduced from i. 6, if wanted, at any time after we have passed i. 13, which can be used to prove that, if the angles formed by producing two sides of a triangle beyond the base are equal, the base angles themselves are equal. Proclus adds a proof of the converse of the second part of i. 5, Le. of the proposition that, if the angles formed by producing two sides of a triangle beyond the base are equal, the triangle is isosceles; but it runs to some length and then only effects a reduction to the theorem of i. 6 as we have it. As the result of this should hardly be assumed, a better proof would be an independent one adapting Euclid's own method in i. 6. Thus, with the construction of i. 5, we first prove by means of i. 4 that the triangles BFC^ CGB are equal in aU respects, and therefore that FC is equal to GB^ and the angle BFC equal to the angle CGB. Then we have to prove that AF^ AG sure equal. If they are not, let AF be the greater, and from FA cut off FH equal to GA. Join CH. H. E. 17 JS8 hOOlt t ti.6,7 Then we have, in the two triaz^les HFC^ AGB^ two sides HF^ FC equal to two sides AG^ GB and the angle ^/C equal to the angle AGB. Therefore (i. 4) the triangles HFC^ AGB are equal But the triangles BFC, CGB are also equal Therefore (if we take away these equals respectively) the triangles HBQ ACB are equal: which is impossible. Therefore AF^ AG are not unequal. Hence AF\& equal to AG and, if we subtract the equals BF^ CG respec- tively, AB is equal to AC. This proof is found in the commentary of an-Naiibl (ed. Besthom-Heibeigi p. 61 ; ed. Curtze, p. 50). Alternative proofs of I. 6. Todhunter points out that i. 6, not being wanted till 11. 4, could be postponed till later and proved by means of i. 26. Bisect the angle BAC by a straight line meeting the base at 2>. Then die triangles ABD^ ACD are equal in all respects. Another method depending on i« 26 is given by an-Nairld after that proposition. Measure equal lengths BD^ CE along the sides BA^ CA. Join BE, CD. Then [i. 4] the triangles DBC, ECB are equal in all respects; therefore £B, DC are equal, and the angles BEC, CDB are equal. The supplements of the latter angles are equal [1. 13], and hence the triangles ABE, ACD luive two angles equal respectively and the side BE equal to the side CD. Therefore [1. 26] AB is equal to AC. Proposition 7. Given two straight, lines constructed on a straight line {from its extremities) and fneeting in a point, there cannot be constructed on the same straight line {from its extremities\ and on the same side of it, two other straight lines meeting tn 5 another point and equal to the former two respectively, namely each to that which has the same extremity with it. For, if possible, given t^o straight lines AC, CB con- structed on the straight line AB and meeting at the point C, let two other straight lines 10 AD, DB be constructed on the same straight line AB, on the same side of it, meeting in another point D and equal to the former two respectively, namely each to that which has the same extremity with it, so that CA is 15 equal to DA which has the same extremity A with it, and ,1 L 7] PROPOSITIONS 6, 7 ^59 CB to DB which has the same extremity B with it ; and let CD be joined. Then, since -^C is equal to ADy the angle ACD is also equal to the angle ADC; [i. s] therefore the angle ADC is greater than the angle DCB ; therefore the angle CDB is much greater than the ancfle DCB. \ Again, since CB is equal to DB, " \ the angle CDB is also equal to the angle DCB. ■ |; But it was also proved much greater than it : which is impossible. Therefore etc. q. e. d. i! 1—6. In an English translation of the enunciation of this proposition it is absolutely "...... . wnicn . . - I The reason is partly that the Greek enunciation is itself very elliptioU, and partly that some words used in it conveyed more meaning than the corresponding words in English do. Particularly is this the case with 06 ffvera^orrai M "there shall not be constructed upon," since ^wi^rmff9ai is the regular word for constructing a triangle in particular. Thus a Ureek ' would easily understand vwraBi/faoirrai 4vl as meaning the construction of two Unts forming' a triangle on a given straight line as base ; whereas to "construct two straight lines on a straight line*' is not in English sufficiently de6nite unless we explain that they are drawn from the ends of the straight line to meet at a point. I have had the less hesitation in putting in the words "from its extremities" because they are actually used by Euclid in the somewhat similar enunciation of I. 91. How impossible a literal translation into English is, if it is to convey the meaning of the enunciation mtelligibly, will be clear from the following attempt to render literally: **On the same straight line there shall not be constructed two other straight lines equal, each to each, to the same two straight lines, (terminatixig) at different points on the same side, having the same extremities as the original straight lines *' (M r^f 0^1% tMtlat M rsut adroit eiStUut \tidp^ tA oMl Wpara ix^wm rtus ^ d^X^t ci^^eiait). i The reason whv Euclid allowed himself to use, in this enunciation, language apparently I so obscure is no doubt that the phraseology was traditional and therefore, vague as it was, I had a conventional meaning whidi the contemporary geometer well understood. This is I proved, I think, by the occurrence in Aristotle {Metearvlogiea lii. 5, 376 a 2 sqq.) of the very 'same, evidently technical, expressions. Aristotle is there alluding to the theorem given by j Entodus from Apollonius' Plane Loci to the effect that, if /T, ^ be two fixed points and M such a variable point that the ratio of MH to MK is a given ratio (not one of equality^ the locus of A/ is a circle. (For an account of this theorem see note on vi. 3 below.) Now Aristotle says " The lines drawn up from /^, AT in this ratio cannot be constructed to two different points of the semicircle ^ ** (ol oj^r drd rQnf HK dMiY^/««yai ypoLtiftaX ^ rodry r^ Xiytf oi ovvTaSiiVQwrai roO i^l', f A iifUKWcktov vpbt AWo Kal dXXo ffii/ittow). If a paraphrase is allowed instead of a translation adhering as closely as possible to the original, Simson's is the best that could be found, since the fiiict that the straight lines form triimgles on the same base is really conveyed in the Greek. Simson*s enunciation is. Upon the same Aau, and on the same side of it^ there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another^ and likewise those vihieh art terminated at the other extremity, Th. Taylor (the translator of Produs) attacks 'imson's alteration as "indiscreet" and as detractmg from the beauty and accuracy of Jodid's enunciation which are enlarged upon by Proclus in his commentary. Yet, when Taylor says ** Whatever difficulty learners may find in conceiving this proposition abstractedly b CMily removed by its exposition in the figure," he really gives his case away. The fact is l|that Taylor, always enthusiastic over his author,, was nettled by Simson's sighting remarks on Proclus' comments on the proposition. Simson had said, with reference to Proclus' explanation of the bearing of the second part of i. 5 on i. 7, that it was not "worth while 17 — 2 36o fiook t [1.7 to relate his trifles mt full length," to whidi Taylor retorts ''But Mr Simaoo was no philosopher ; and therefore the greatest part of these Commentaries must be oonsiderad fay nim as trifles, from the want of a phuoaophic genius to comprehend their meaning, and a taste superior to that of a nc/rv mmthmtUidam^ to discover their beauty and d^guoe.** so. It would be natural to insert here the step '*lmt the angle ACD is greater than the angle ^C/>. [C. A^. 5.]" 1 1 . much greater, literally " greater by much " (voXXf /mI^) • Simsoo and thoae who follow him transUte : ^^much mwn tkm is the angfle BDC grmitr than the anrie BCD^^ but the Greek for this would have tobe veXX^ (or a-oXv) ^kWkw ^#n...#Mf^. nXXf fJOJm^ however, though used by Apollonius, Is not, apparently, found in Endid or Archimedes. Just as in 1. 6 we need a Postulate to justify tbeoreticaQy die statement that CD falls within the angle ACB^ so that the triangle DBC is less than the triangle ABQ so here we need Postulates which shall satisfy us as to the relative positions of CA^ CB^ CD on the one hand and of />C, DA^ DB on the other, in order that we may be able to infer that the angle BDC is greater than the angle ADC^ and the angle ACD greater than the angle BCD. De Morgan {op. at. p. 7) observes that 1. 7 would be made easy to beginners if they were first familiarised, as a common notion, with **if two magnitudes be equal, any magnitude greater than the one is greater than any magnitude less than the other." I doubt however whether a beginner would follow this easily ; perhaps it would be more easily apprehended in the form ''if any magnitude A is greater than a magnitude B^ the magnitude A is (a fortiori) greater than any magnitude less than B or than any magnitudie equal to B." It has been mentioned already (note on i. 5) that the second case of l 7 given by Simson and in our text-books generally is not in the original text (the omission being in accordance with Euclid's general practice of giving only one case, and tiiat the most difficult, and leaving the others to be worked out by the reader for himself). The second case is given by Produs as the answer to a possible objection to Euclid's proposition, which should assert that the proposition is not proved to be universally true, since the proof given does not cover all possible cases. Here the objector is supposed to contend that what Euclid declares to be impossible may still be possible if one pair of lines lie wholly within the other pair of lines; and the second part of i. 5 enables the objection to be refuted. If possible, let AD^ DB be entirely within the triangle formed by AC^ CB with ABy and let AC be equal to AD and BC \ to BD. Join CD^ and produce AC^ AD to E and F. Then, since AC is equal to AD^ the triangle ACD is isosceles, and the angles ECD^ FDC under the base are equal. But the angle ECD is greater than the angle BCD \ therefore the angle FDC is also greater than the angle BCD. Therefore the angle BDC is greater by far than the angle BCD. Again, since DB is equal to CB^ the angles at the base of the triangle BDC are equal, [i. 5 ^ that is, the angle BDC is equal to the angle BCD. Therefore the same angle BDC is both greater than and equal to the ai^le BCD: which is impossible. The case in which D CeJIs on AC or BC does not require proof. i I. 7, 8] PROPOSITIONS 7, 8 261 I have already referred (note on i. i) to the mistake made by those editors who regard i. ^ as being of no use except to prove i. 8. What i. 7 proves is that if[ in addition to the base of a triangle, the length of the side terminating at each extremity of the base is given, only one triangle satisfying these conditions can be constructed on one and the same side of the given base. Hence not only does i. 7 enable us to prove i. 8, but it supplements 1. I and I. 22 by showing that the constructions of those propositions ^ve one triangle only on one and the same side of the base. But for i. 7 this could not be proved except by anticipating iii. 10, of which therefore i. 7 is the equivalent for Book i. purposes. Dodgson (Euclid and his modem Rivals^ pp. 194 — 5) puts it in another way. " It [i. 7] shows that, of all plane figures that can be made by hingeing rods together, the thrte-siAeA ones (and these only) are rigid (which is another way of stating the fact that there cannot be iw0 such figures on the same base). This is imalogous to the fact, in relation to solids contained by plane surfaces hinged together, that any such solid is rigid, there being no maximum number of sides. And there is a close analogy between i. 7, 8 and iii. 23, 24. These analogies give to geometry much of its beauty, and I think that they ousht not to be lost sight of." It will therefore be apparent how ill-advised are those editors who eliminate i. 7 altogether and rely on Philo's proof for i. 8. • Proclus, it may be added, gives (pp. 268, 19 — 269, 10) another explanation of the retention of i. 7, notwithstanding that it was apparently only required for I. 8. It was said that astronomers used it to prove that three successive eclipses could not occur at equal intervals of time, i.e. that the third could not follow the second at the same interval as the second followed the first ; and it was ai|;ued that Euclid had an eye to this astronomical application of the proposition. But, as we have seen, there are other grounds for retaining the proposition which are quite sufficient of themselves. Proposition 8. 1/ two triangles have the two sides equal to two sides respectively, and have also the base equal to t/ie base, they will aba have the angles equal which are contained by the equal straight lines. \ Let ABC, DEF be two triangles having the two sides \AB, AC equal to the two sides ^DE, DF respectively, namely .AB to DE, and AC to DF\ and ^let them have the base BC equal to the base EF\ 1 say that the angle BAC is also equal to the angle EDF. For, if the triangle ABC be applied to the triangle DEF, and if the point B be placed on [the point E and the straight line BC on EF, the point C will also coincide with F, because BC is equal to EF. 263 BOOK I [l8 Then, BC coinciding with EF^ BA. AC will also coincide with ED, DF\ » for, if the base BC coincides with the base EF, and the sides BA, AC do not coincide with ED, DF but fall beside them as EG, GF, then, given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there will sshave been constructed on the same straight line (from its extremities^, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it 30 But they cannot be so constructed. [i. 7] Therefore it is not possible that, if the base BC be applied to the base EF, the sides BA, AC should not coincide with ED, DF; they will therefore coincide, 35 so that the angle BAC will also coincide with the angle EDF, and will be equal to it If therefore etc. q. e. d. 19. B A, AC. The text has here " BA, CA, " 91. fall beside them. The Greek has the future, wa^taXk^qvvi, wapoXKirrm means " to pass by without touchiug,** ** to miss" or *' to deviate." As pointed out above (p. 257) i. 8 is a /or/Za/ converse of i. 4. It is to be observed that in i. 8 Euclid is satisfied with proving the equality of the vertical angles and does not, as in i. 4, add that the triangles are equal, and the remaining angles are equal respectively. The reason is no doubt (as pointed out by Proclus and by Savile afler him) that, when once the vertical angles are proved equal, the rest follows from i. 4, and there is no object in proving again what has been proved already. Anstotle has an allusion to the theorem of this proposition in MeteorologUa III. 3, 373 a 5 — 16. He is speaking of the rainbow and observes that, if equal ra3rs be reflected from one and the same point to one and the same point, the points at which reflection takes place are on the circumference of a circle. "For let the broken lines ACB, AFB, ADB be all reflected from the point A to the point B (in such a way that) AC, AF, AD are all equal to one another, and the lines (terminating) at B, Le. CB, FB, DB, are likewise all equal ; and let AEB be joined. It follows that the triangles are equal; for they are upon the equal (base) AEB*^ Heibe^ {Mathematisches nt Aristoteles, p. 18) thinks that the form of the conclusion quoted is an indication that in the corresponding proposition to Eud. I. 8, as it lay before Aristotle, it was maintained that the triangles were equal, and not only the angles, and "we see here therefore, in a clear example, how the stones of the ancient fabric were recut for the rigid structure of his 18] PROPOSITION 8 363 I I Elements.^ I do not, however, think that this inference from Aristotle's language as to the form of the pre-Euclidean proposition is safe. Thus if we, nowadays, were arguing from the data in the passage of Aristotle, we should doubtless infer dir^y that the triangles are equal in all respects, quoting i. 8 alone. Besides, Aristotle's language is rather careless, as the next sentences of the same passage show. " Let perpendiculars," he says, *'be drawn to AEB from the angles, CE from C, FE from ^and DE from D. These, then, are equal j for they are all in equal triangles, and in one plane; for all of them are perpendicular to AEB^ and they meet at one point E. There- fore the (line) drawn (through C, F, 2P) will be a circle, and its centre (will be) E'* Aristotle should obviously have proved that the three perpendiculars will meet at one point E on AEB before he spoke of drawing the perpendiculars CE^ FE, DE, This of course follows from their being " in equal triangles " (by means of EucL I. 26); and then, from the fact that the perpendiculars meet at one point on AB, it can be inferred that all three are m one plane. Philo'8 proof of I. 8. This alternative proof avoids the use of i. 7, and it is elegant ; but it is inconvenient in one respect, since three cases have to be distinguished. Proclus gives the proof in the following order (pp. 266, 15 — 268, 14). I^ ABC, DEF be two triangles having the sides A By -<4C equal to the sides DEy 2?^ respectively, and the base ^C equal to the base EF, Let the triangle ABC be applied to the triangle DEF, so that B is placed on E and BC on EF, but so that A falls on the opposite side of EF from D, taking the position G. Then C will coincide with F, since BC is equal to EF. Now FG will either be in a straight line with DF, or make an angle with it, and in the latter case the angle will either be interior (icara to ivro^) to the figure or exterior (icara ro licw). I. Let FG he in a straight line with DF Then, since DE is equal to EG, and DFG is a straight line, DEG is an isosceles triangle, and the angle at Z^ is equal to the angle at G. [I- s]. II. Let DFy FG form an angle interior to the figure Let DG be joined. Then, since DE, EG are equal, the angle EDG is equal to the angle EGD. Again, since DF\& equal to FG, the angle FDG is equal to the angle FGD. Therefore, by addition, the whole angle EDF is equal to the whole angle EGF, 364 BOOK I [1.8, n III. Let DF^ FG form an angle txterwr to the figure. Let DG be joined. The proof poceeds as in the last case, except that subtraction takes the place of addition, and the remaining angle EDF is equal to the remaining angle EOF. Therefore in all three cases the angle EDF is equal to the angle EGF^ that is, to the angle >ff^C. It will be observed that, in accordance widi the practice of the Greek geometers in not recognising as an "angle " aiw angle not less than two right angles, the re-entrant angle of the quadrilateral JDEuF\% ignored and the ang^ DFG is said to be outside the figure. Proposition 9. To bisect a given rectilineal angle. Let the angle BAC be the ^ven rectilineal angle. It is then required to bisect it. Let a point D be taken at random on AB ; let AE be cut off from AC equal to AD ; [i. 3] let DE be joined, and on DE let the equilateral triangle DEFh^ constructed ; let .^/^ be joined. I say that the angle BAC has been bisected by the straight line AF. r or, since AD is equal to AE, and AF is common, the two sides DA, AF are equal to the two sides ^ EA, AF respectively. And the base DF is equal to the base EF\ therefore the angle DAF is equal to the angle EAF. j [I. 8] I Therefore the given rectilineal angle BAC has been | bisected by the straight line AF. Q. e. f. ^ It win be observed from the translation of this proposition that Euclid does not say, in his description of the construction, that the equilateral triangle | should be constructed on the side of DE opposite to ^ ; he leaves this to be < inferred from his figure. There is no particular value in Proclus' explanation \ as to how we should proceed in case any one should assert that he could not . recognise the existence of any space below DE. He supposes, then, the ! equilateral triangle described on the side of DE towards A, and hence has to -. consider three cases according as the vertex of the equilateral triangle fidls on A, above A or below it The second and third cases do not difier i I. 9] PROPOSITIONS 8, 9 265 substantially from Euclid's. In the first case, where ADE is the equilateral triangle constructed on DE^ take any point ^on AD^ and from AE cut off AG equal to AF, Join DG^ J?^ meeting in H\ and join AH, Then ^^ is the bisector required. Proclus ako answers the possible AjecHon that might be raised to Euclid's proof on the ground that it assumes that, if the equilateral triangle m described on the side of DE opposite to A^ its vertex -F will lie within the angle BAC. The objector is supposed to argue that this is not necessary, but that ^ might fall eiSier on one of the lines forming the angle or outside it altogether. The two cases are disposed of thus. Suppose ^to fall as shown in the two figures below respectively. Then, since FD is equal to FE^ the angle FDE is equal to the angle FED, Therefore the angle CED is greater than the angle FDE ; and, in the second figure, a fortiori^ the angle CED is greater than the angle BDE. But, since ADE is an isosceles triangle, and the equal sides are produced. the angles under the base are equal, i.e., the angle CED is equal to the angle BDE, But the an^e CED was proved greater : which is impossible. Here then is the second case in which, in Proclus' view, the second part of I. 5 is useful for refuting objections. On this proposition Proclus takes occasion (p. 371, 15—19) to emphasize I the isjct that the given angle must be rectilineal^ since the bisection of any sort of angle (including angles made by curves with one another or with straight \ lines) is not matter for an elementary treatise, besides which it is questionable ^whether such bisection is always possible. "Thus it is difficult to say iwhether it is possible to bisect the so-called horn-like angle " (formed by the prcumference of a circle and a tangent to it). Trisection of an angle. Further it is here that Proclus gives us his valuable historical note about ithe trisection of any acute angle, which (as well as the division of an angle in iny ^ven ratio) requires resort to other curves than circles, i.e. curves of the speaes which, after Geminus, he calls "mixed." "This," he says (p. 372, I — 12), "is shown by those who have set themselves the task of trisecting such a given rectilineal angle. For Nicomedes trisected any rectilineal anple by means of the conchoidal lines, the origin, order, and properties of iduch he has handed down to us, being himself the discoverer of their peculiarity. Others have done the same thing by means of the quadratrices of Hippias ..and Nicomedes, thereby again using 'mixed' curves. ^ But others, starting I' from the Archimedean spirals, cut a given rectilineal angle in a given ratio." * ■ r 366 -JLJLl "•"■■> BOOK I [1.9 (a) Trisection by means of the amckaid. I have already spoken of the ionchmd of Nioomedes (nole on Det a, pp. 160 — i) ; it remains to show how it could be nsed for trisecting an angle. Pappus explains this (iv. pp. 374 — 5) as follows. Let ABC be the given acute angle, and from any point A in AB draw A C perpendicular to BC. e O Complete the parallelogram FBCA and produce FA \xi z, point E luch that, if BE be joined, BE interapis between AC and AE a Ungtk DE fual to twice AB. I say that the angle BBC is one-third of the angle ABC For, joining A to G, the middle point of DE^ we have the three straUht . lines AG, DG, EG equal, and the angle AGD is double of the angle AED or EBC But DE is double of AB ; therefore AG, which is equal to DG, is equal to AB. Hence the angle AGD is equal to the angle ABG. Therefore the angle ABD is also double of the angle EBC; so that the angle EBC is one-third of the angle ABC, So bi Pappus, who reduces the construction to the drawing of BE so that DE shall be equal to twice AB. This is what the conchoid constructed with Bz&pole, AC2& directrix^ and distance equal to twice AB enables us to do ; for that conchoid cuts AE in the required point E. {b) Use of the quadratrix. The plural quadratrices in the above passage is a Hellenism for the singular quadratrix, which was a curve discovered by Hippias of Elis about 4^ B.C. According to Produs (p. 356, 11) Hippias proved its properties ; and we are told (i) in the passage quoted above that Nicomedes also investi^ted it and Uiat it was used for trisecdi^ an angle, and (3) by Pappus (iv. pp. 250, 33 — 252, 4) that it was used by Dmostratus and Nicomedes and some more recent writers for squaring the circle, whence its name. It is described thus (Pappus iv. p. 252). Suppose that ABCD is a square and BED a quadrant of a circle with centred. Suppose (i) that a radius of the circle moves uniformly about A from the position AB to the position AD, and (2) that in the same time the line BC moves uniformly, always parallel to itself, and with its jextremity B moviiig along BA, from the position BC to the position AD. Then the radius AE and the moving line BC determine at any instant by their intersection a point F. The locus of ^is the quadratrix. p l '^ V ♦ 1 1 !iQ i > , I. 9, lo] PROPOSITIONS 9, 10 267 1 The property of the curve is that, if ^ is any point, the arc BED is • to the arc ED as AB is to FH. i In other words, if ^ is the angle FAD^ p the radius vector AFand a the aide of the square, (p8in^)/a = ^/Jir. ^ Now the angle EAD can not only be trisected but divided in any git*en I ratio by means of the quadratrix (Pappus iv. p. 286). I Fof let FH be divided at K in the given ratio. I Draw KL parallel to AD^ meeting the curve in L ; join AL and produce I it to meet the circle in N. Then the angles EAN^ NAD are in the ratio of FK to KH^ as is easily proved. (c) Use of the spiral of Archimedes. The trisection of an angle, or the division of an angle in any ratio, by means of the spiral of Archimedes is of course an equally simple matter. Suppose any angle included between the two radii vectores OA and OB of the ^^Md, and let it be required to cut the angle AOB in a given ratio. Since the radius vector increases proportionally with the angle described by the sector which generates the curve (reckoned from the original position of the (rector coinciding with the initial line to the particular position assiuned), we iiave only to take the radius vector OB (the greater of the two OA^ 0B\ nark off OC along it equal to OA^ cut CB in the given ratio (at D say), and fhen draw the circle with centre O and radius OD cutting the spiral in E, uhen OE will divide the angle AOB in the required manner. I Proposition id. To bisect a given finite straight line. Let AB be the given finite straight line. Thus It is required to bisect the finite straight line AB. Let the equilateral triangle ABC be .K>nstructed on it, [i. i] ind let the angle ACB be bisected by the traight line CD ; [1. 9] \ I say that the straight line AB has ■•en bisected at the point D. - For, since ACv& equal to CB, id CD is common, I the two sides AC, CD are equal to the two sides Bd 'Z? respectively ; nd the angle A CD is equal to the angle BCD ; \ therefore the base AD is equal to the base BD. [i. 4} iTherefore the given finite straight line AB has been lisected at D. q. e. f. 268 BOOK I [I. lol [1.8; Apollonius, we are told (Produs, pp. 279, 16— 280, 4), bisected a stimight line AB by a construction like that of i. i. With centres A^ B, and raiUi AB^ BA respec- tively, two circles are described, intersecting in C, D. ]o\nmg CD, AC, CB, AD, DB, Apol- lonius proves in two steps that CD bisects AB. (i) Since, in the triangles A CD, BCD, two sides AC, CD are equal to two sides BC, CD, and the bases AD, BD are equal, the angle A CD is equal to the angle BCD. (2) The latter angles being equal, and AC being equal to CB^ while C£ is common, the equality of AE, EB follows by l 4. The objection to this proof is that, instead of assuming the bisection o the angle ACB, as already effected by i. ^ ApoUonius goes a step fiuthe» back and embodies a construction for bisecting the ang^ That iS| bt unnecessarily does over again what has been done before, which is open to objection from a theoretiad point of view. Proclus (pp. 277, 25 — 279, 4) warns us against being moved by thisi proposition to conclude that geometers assumed, as a preliminary hypothesis. | that a line is not made up of indivisible parts (If Vip«r). This might bt| argued thus. If a line is made up of indivisibles, there must be in a finitti line either an odd or an even number of them. If the number were odd,! it would be necessary in order to bisect the line to bisect an indivisible (the) odd one). In that case therefore it would not be possible to bisect a straight line, if it is a magnitude made up of indivisibles. But, if it is not so made up, the straight line can be divided ad infinitum or without limit (hr cnrcipor Staipciroi). Hence it was argued (^oo-u'), says Proclus, that the divisibility of magnitudes without limit was admitted and assumed as a geometrical principle. To this he replies, following (jeminus, that geometers did indeeci assume, by way of a common notion, that a continuous magnitude, Le. magnitude consisting of parts connected together (ounififMHtfr), is diviribli (8uup«Tov). But infinite divisibility was not assumed by them ; it was pravei by means of the first principles applicable to the case. **For when," h sa3rs, ''they prove that the incommensurable exists among magnitudes, an that it is not all things that are commensurable with one another, wlu else will any one say that they prove but that every ma^tude can \ divided for ever, and that we sludl never arrive at the mdivisible, tb* is, the least common measiure of the magnitudes? This then is matto* . demonstration, whereas it is an axiom that everything continuous is divisible so that a finite continuous line is divisible. The writer of the Elemeni bisects a finite straight line, starting from the latter notion, and not fix»m an assumption that it is divisible without limit ** Proclus adds that the propositio may also serve to refute Xenocrates' theory of indivisible lines (aro^ ypofifjuai The aigument given by Proclus to disprove the existence of indivisible linr is substantially that used by Aristotle as r^;ards magnitudes generally (c PAysics VI. I, 231 a 21 sqq. and especially vi. 2, 233 b 15 — 32). Ill] PROPOSITIONS lo, II 269 Proposition ii. I To draw a straight line at right angles to a given straight line front a given point on it. Let AB be the given straight line, and C the given point on it. Thus it is required to draw from the point C a straight . line at right angles to the straight \ line AB. I Let a point D be taken at ran- \ dom on AC\ ^ let CE be made equal to CD ; [i. 3] ► on DE let the equilateral triangle \FDE be constructed, • [i. i] I and let FC be joined ; I say that the straight line FC has been drawn at right I angles to the given straight line AB from C the given point Jon it. • For, since DC is equal to CE, rCF is common, the two sides DC, CF are equal to the two sides EC, I ' CF respectively ; I and the base DF is equal to the base FE ; therefore the angle DCF is equal to the angle ECF\ ['• «] and they are adjacent angles. But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal ingles is right ; [Def. 10] therefore each of the angles DCF^ FCE is right Therefore the straight line CF has been drawn at right ngles to the given straight line AB from the given point r on it. Q. E. F. 10. let CE be made equal to CD. The verb is ntlrSw which, as well as the other barts of ccifuu, is constantly used for the passive of rWriiu " to fiiace^ ; and the latter word is constantly used in the sense of making, e.g., one straight line equal to another straight line. De Morgan remarks that this proposition, which is " to bisect the angle inade by a straight line and its continuation " [Le. a flat angle], should be a particular case of i. 9, the constructions being the same. Tms is certainly 270 BOOK I [i. lit IS worth noting, though I doubt the advantage of rearranging the propositions in consequence. Apollonius gave a construction for this proposition (see Proclus, p. 282, 8) differing from Euclid's in much the same way as his construction for bisecting a straight line differed from that of i. 10. Instead of assuming an equilatenl triangle drawn without repeating the process of i. i, Apollonius takes D and E equidistant from C as in Euclid, and then draws circles in the manner of I. I meeting at F. This necessitates proving again that DFv^ equal to FE\ whereas Euclid's assumption of the construction of i. i in the wc^ *' let the equilateral triangle FDE be constructed " enables him to dispense with the drawing of circles and with the proof that DF is equal to FE at the same time. While however the substitution of Apollonius* constructions for l 10 and II would show faulty arrangement in a theoretical treatise like Eudid's, they are entirely suitable for what we call practical geometry, and such may have been Apollonius' object in these constructions and in his alternative for I. 23. Proclus f^ves a construction for drawing a straight line at right angles to another straight line but from one end of it, instead of from an intermediate point on it, it being supposed (for the sake of argument) that we are not permitted to produce the straight line In the commentary of an-NaiiM (ed. Besthom-Heiberg, pp. 73 — 4; ed. Curtze, pp. 54 — 5) this constructiv attributed to Heron. Let it be required to draw from A a straight line at right angles to AB. . On AB take any point C, and in the manner of the proposition draw CE^ at right angles to AB, From CE cut off CD equal to AC^ bisect the angle ACE by the straight line CF^ [i. 9] and draw DF at right angles to CE meeting CF in F Join FA. Then the angle FAC will be a right angle. For, since, in the triangles ACF^ DCF^ the two sides AC^ CF are equal to the two sides DC^ CF respectively, and the included angles ACF^ DCFzx^ equal, the triangles are equal in all respects. [i. 4 Therefore the angle at A is equal to the angle at Z>, and is accordingly ' right angle. E F D \ A < \ 6 Proposition 12. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. Let AB be the given infinite straight line, and C the given point which is not on it ; t I I. 12] PROPOSITIONS II, 12 271 5 thus it is required to draw to the given infinite straight line AB^ from the given point C which is not on it, a per- pendicular straight line. For let a point D be taken 10 at random on the other side of the straight line AB, and with centre c and distance CD let the circle EFG be described ; [Post 3] let the straight line EG IS be bisected at H, [i. 10] and let the straight lines CG, CH, CE be joined. [Post l] I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it. 20 For, since GH is equal to HE^ and HC is common, flie two sides GH, HC are equal to the two sides EH, HC respectively ; and the base CG is equal to the base CE ; 25 therefore the angle CHG is equal to the angle EHC. [1.8] And they are adjacent angles. But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is 30 called a perpendicular to that on which it stands. [Def. 10] Therefore C/^ has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it Q. E. F. 3. a perpendicular straight line, KdBrrm edtfciov ypaf»M^p, This is the full expression -for a perpiHdicular, ttiBrrw meaning lei faU or let dawn^ so that the expression corresponds to our plumb-line, ^ jrdtfrrot b however constantly used alone for a perpendicular, ypofifiilj being understood. ID. on the other side of the straight line AB, literally '* towards the other parts of the straight line AB,'' iwl rd Irfpa fU/ni r^t AB. Cf. **on the same side" (M rd o^d ftdfni) in Post. 5 and **in both directions" (^* ixdnpa rd fU/ni) in Def. 33. ** This problem,*' says Produs (p. 283, 7—10), "was first investigated by Oenopides [5th cent b.c], who thought it useful for astronomy. He however calls the perpendicular, in the archaic manner^ (a line drawn) 272 BOOK I [i. 12 gmmum-wise (mra yr«fuyvaX because the gnomon is also at right anf^es to the horizon." In this earlier sense the gnomon was a staff {daoed in a vertical position for the purpose of casting shadows and so serving as a means of measuring time (Cantor, GesckickU der Mathimatik^ i,, p. 161). The later meanings of the word as used in Eucl. Book 11. and elsewhere will Le explained in the note on Book lu Def. 2. Proclus says that two kinds of perpendicular were distinguished^ the ''plane" (IrtircSof) and the "solid" (orfpco), the former being the perpendicular dropped on a line in a plane and the latter theperpenmcular dropped on a plane. The term "solid perpendicular" is sufficiently curious, but it may perhaps be compared with the Greek term ''solid locus" applied to a conic section, apparently on the ground that it has its origin in the section of a solid, namely a cone. Attention is called by most editors to the assumption in this proposition that, if only D be taken on the side oi AB remote from C, the circle described with CD as radius must necessarily cut A£ in two points. To sati^ us of this we need, as in i. i, some postulate of continuity, e.^. something like that suggested by Killing (see note on the Principle of Contmuity above, pu 235}: " If a point [here the point describing the arde] moves in a figure which is divided into two parts [by the straight line], and if it belongs at the beginning of the motion to one part and at another stage of the motion to the other ps^ it must during the motion cut the boundary between the two parts," and this of course applies to the motion in iufo directions fix>m D. But the editors have not, as a rul^ noticed a possible ctjecHon to the Euclidean statement of this problem which is much more difficult to dispose of at this stage, i.e. without employing any proposition later than this in Euclid's order. How do we know, says the supposed critic, that the cirde does not cut AB in thra or more points, in which case there would be not one perpendicular but three or more? Proclus (pp. 286, 12—289, 6) tries to refute this objection, and it is interesting to follow his argument, though it will easily be seen to be inconclusive. He takes in order three possible suppositions. I. May not the circle meet AB in a third point K between the middle point of GE and either extremity of it, taking the form drawn in the figure appended ? Suppose this possible. Bisect GE in H. Join CJ7, and produce it to • meet the circle in Z. Join CG^ CK, CE. Then, since CG is equal to CE^ and CH is common, while the base GH is equal to the base HE^ the angles CHG^ CHE are equal and, since they are adjacent, they are both right Again, since CG is equal to CE^ the angles at G and E are equal. Lastly, since CK is equal to CG and also to CE, the angles CGK, CKG are equal, as also are the angles CKE, CEK. Since the angles CGK, CEJCare equal, it follows that the angles CJCG, CITE are equal and therefore both right. Therefore the angle CKHis equal to the angle CHK, and CH\& equal to CK. I. 12] PROPOSITION 12 273 But CK is equal to CZ, by the definition of the circle ; therefore CH is equal to CL : which is impossible. Thus Proclus; but why should not the circle meet AB m H2s well as JT? 2. May not the circle meet AB in H the middle point of GE and take the form shown in the second fi^re? In that case, says Proclus, jom CG^ CHy CE as before. Then bisect HE at iT, join CK and produce it to meet the circumference at 2. Now, since HK is equal to KE^ CK is common, and the base CH is equal to the base CE^ the angles at K are equal and therefore both right angles. Therefore the angle CHK is equal to the angle CKH^ whence CKv& equal to CH and therefore to CL\ which is impossible. So Proclus ; but why should not the circle meet AB in JT as well as /T? 3. May not the circle meet AB in two points besides G^ E and pass, between those two points, to the side of AB towards C, as in the next figure ? Here again, b^ the same method, Proclus proves that, K^ L being the other two points m which the circle cuts AB, CK is equal to CH, and, since the circle cuts CH in M, CM is equal to CK and therefore to tCH\ which is impossible. But, again, why should the circle not \ cut AB in the point Hz& well? In &ct, Proclus' cases are not mutually exclusive, and his method of proof only enables us to show that, if the circle meets AB in one more point besides G^ E^ it must meet it in more points still We can always find a new point of intersection by bisecting the distance separating any two points of intersection, and so, applying the method ad infinitum, we should have to conclude ultimately that the circle with radius CH (or CG) coincides with AB, It would follow that a drcle with centre C and radius greater than CH would not meet AB at all. Also, since all straight lines from C to points on AB would be equal in length, there would be an infinite number of perpendiculars from C on AB. Is this under any circumstances possible? It is not possible in Euclidean space, but it is possible, under the Riemann hypothesis (where a straight line is a "closed series" and returns on itself), in the case where C is the pole of the straight line AB. It is natural therefore that, for a proof that in Euclidean space there is only one perpendicular firom a point to a straight line, we have to wait until 1. 16, the precise proposition which under the Riemann hypothesis is only valid with a certain restriction and not universally. There is no difficulty involved by waiting until i. 16, since i. 12 is not used before that proposition is reached ; and we are only in the same position as when, in order to satisfy ourselves of the number of possible solutions of i. i, we have to wait till i. 7. I But if we wish, after all, to prove the truth of the assumption without recourse to any later proposition than i. 12, we can do so by means of this same invaluable i. 7. H. E. 18 «74 BOOK I If the circle intersects AB as before in tr, jE^ let i7be the middle point of GE^ and suppose, if possible, that the circle also intersects AB in any other point AT on ^^. From H^ on the side of AB opposite to C, draw HL at right angles to AB^ and make HL equal to HC. Join CG, LG, CK, LK. Now, in the tnangles CBG^ LHG^ CH'xs equal to LH^ and HG is common. Also the angles CHG, LHG^ being both right, are equal. Therefore the base CG is equal to the base LG. Similarly we prove that CK is equal to LK. But, by hypothesis, since A* is on the circle, CK\& equal to CG. Therefore CG, CK, LG, LKzxe all equal. Now the next proposition, i. 13, will tell us that Clf, HL are in a straight line; but we will not assume this. Join CZ. Then on the same base CL and on the same side of it we have two pairs of straight lines drawn from C, L to G and K such that CG b equal to CK and LG to LK But this is impossible [i. 7]. Therefore the circle cannot cut BA or BA produced in any point other than G on that side of CL on which G is. Similarly it cannot cut AB or AB produced at any point other than E on the other side of CL> The only possibility left therefore is that the circle might cut AB in the same point as that in which CL cuts it But this is shown to be impossible by an adaptation of the proof of i. 7. For the assumption is that there may be some point M on CL such that CM is equal to CG and ZAT to LG. If possible, let this be the case, and produce CG toN. Then, since CM is equal to CG, the angle NGM is equal to the angle GML [i. 5, part 2]. Therefore the angle GML is greater than the angle MGL. Again, since LG is equal to LM, the angle GML is equal to the angle MGL. But it was also greater : which is impossible. Hence the circle in the original figure cannot cut AB in the point in which CL cuts it Therefore the circle cannot cut AB xn any point whatever except G and E. [This proof of course does not prove that CK is less than CG, but only that it is not equal to it The proposition that, of the obliques drawn from C to AB, that is Jess the foot of which is nearer to ZTcan only be proved later. The proof by i. 7 also fails, under the Riemann hypothesis, if C, Z are the poles of the straight line AB, since the broken lines CGL, CKL etc. become equal straight lines, all perpendicular to AB\ Produs rightly adds (p. 289, 18 sqq.) that it is not neassary to take D on the side of AB away from A^iwa objector ''says that there is no space on I. 12, 13] PROPOSITIONS 12, 13 275 that side." If it is not desired to trespass on that side of AB^ we can take D anywhere on AB and describe the €irc of a circle between D and the point where it meets AB again, drawing the arc on the side of AB on which C is. If it should happen that the selected point D is such that the circle only meets AB in one pomt {D itself), we have only to describe the circle with CD as radius, then, if ^ be a point on this circle, take /*a point further from C than E is, and describe with CF as radius the circular arc meeting AB in two points. Proposition 13. If a straight line set up an a straight line make angles^ it will make either two right angles or angles equal to two right angles. \ For let any straight line AB set up on the straight line s CD make the angles CBA, ABD ; I say that the angles CBA, ABD are either two right angles or equal to two right angles. Now, if the angle CBA is equal to g to the angle ABD, they are two right angles. [Def. 10] But, if not, let BE be drawn from the point B at right ' angles to CD\ [i. n] therefore the angles CBE, EBD are two right angles, fis Then, since the angle CBE is equal to the two angles CBA, ABE, let the angle EBD be added to each ; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. , [C. N. 2] Again, since the angle DBA is equal to the two angles DBE, EBA, let the angle ABC be added to each ; therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC. [C N. 2] i But the angles CBE, EBD were also proved equal to the same three angles ; and things which are equal to the same thing are also equal to one another ; [C. N. 1] therefore the angles CBE, EBD are also equal to the .0 angles DBA, ABC 18—2 276 BOOK I [h 13, 14 But the angles CBB, EBD are two right angles ; therefore the angles DBA, ABC are also equal to two right angles. Therefore etc. Q. E. D. 17. let the angle EBD be added to each, literally ** let the angle EBD be added (so as to be) common,'* K9Uf^ wfnontiffBu ^ W6 BBA. Sinularly irouH^ d^g^^ is med of subtracting a straight line or angle from each of two others. "Let the common angle EBD be added '^is clearly an inaccurate translation, for the angle is not common before it is added, i.e. the irouH^ is proleptic. ** Let the common angle be sttkraOad** as a translation of atu^ d^pf^Bw woula be less unsatisfactory, it is true, out, as it is desirable to use correraondhig words when translating the two expressions, it seems hopeless to attempt to keep toe won « common,'* and I have therefore said ** to each " and " nrom each " simpfy. Proposition 14. If with any straight line, and at a point on it, two stra^kt lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. 5 For with any straight line AB, and at the point B on it, let the two straight lines BC, BD not lying on the same side make the adjacent angles ABC, ABD equal to two right angles ; I say that BD is in a straight line with CB. o For, if BD is not in a straight line with BC, let BE be in a straight line with CB. Then, since the straight line AB g — stands on the straight line CBE, s the angles ABC, ABE are equal to two right angles. [1. 13] But the angles ABC, ABD are also equal to two right angles ; therefore the angles CBA, ABE are equal to the angles CBA, ABD. [Post 4 and C. N. i] Let the angle CBA be subtracted frpm each ; » therefore the remaining angle ABE is equal to the remaining angle ABD, [C N. 3] the less to the greater : which is impossible. Therefore BE is not in a straight line with CB. Similarly we can prove that neither is any other straight \s line except BD. I. 14, is] propositions 13—15 277 Therefore CB is in a straight line with BD. Therefore etc. ^ « ^ Q. E. D. I. If with any straight line.... There is no greater difficulty in translating the works of the Greek geometers than that of accurately giving the force of prepositions, rp^t, for Instance, is used in all sorts of expressions with various shades of meaning. The present enunciation begins *E^ irpbt ripi tidel^ koI rf *-p6t a&ri ffri/^ff% *nd it is rdly necessary in this one sentence to translate rpdf by three different words, witAt at, and am. The first wp^ must be translated by wit A because two straight lines " makj" an angle vniA one another. On the other hand, where the similar expression wpits rj BodelffTg tdBtl^ occurs in 1. 33, but it is a question of "constructing" an angle (o^uon^ao'tfac), we have to sav "to construct ch a given straight line." Against would perhaps be the English word comine nearest to expressing all these meanings of «7>6f, but it would be intolerable as a tranuation. 17. Todhunter points out that tor the inference in this line Post. 4, that all right angles are equal, is necessary as well as the Common Notion that things which are equal to the same thing (or rather, here, to e^uat tAings) are equal. A similar remark applies to steps in the pro(»s of 1. 15 and i. 38. 34. we can prove. The Greek expresses this by the future of the verb, de(^>/ier, ** we shall prove,'* which however would perhaps be misleading in English. Proclus observes (p. 297) that two straight lines on the same side of another straight line and meeting it in one and the same point may make with one and the same portion of the straight line terminated at the pomt two angles which are together equal to two right angles, in which case however the two straight lines would not be in a straight line with one another. And he quotes from Porphyry a construction for two such straight lines in the particular case where they form with the given straight line angles equd respectively to half a right angle and one and a half right angles. There is no particular value in the construction, which will be gathered from the annexed figure where CE^ CF are drawn at the prescribed inclinations to CD. Proposition 15- If two straight lines cut one another, they make the vertical angles equal to one another. For let the straight lines AB, CD cut one another at the point E ; I say that the angle A EC is equal to ^.^^ the angle DEB, ^^>-^ and the angle CEB to the angle o ^^<~c AED. § For, since the straight line AE stands 3 on the straight line CD, making the angles CEA, AED, the angles CEA, AED are equal to two right angles. [I- 13] 378 BOOK I [l 15 Again, since the straight line DE stands on the straight line AB, making the angles AED\ DEB, the angles AED, DEB are equal to two right angles. IS But the angles CEA, AED were also proved equal to two right angles ; therefore the angles CEA, AED are equal to the angles AED, DEB. * [Post 4 and C A: i] Let the angle AED'h^ subtracted from each ; » therefore the remaining angle CEA is equal to the remaining angle BED. [C. N. 3] Similarly it can be proved that the angles CEB^ DEA are also equal. Therefore etc. q. E. d. 25 [PoRiSM. From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.] I. Uie vertical angles. The diiference between atffacmt angles («l tf^cf^t T wwht) and vertuai angles {ol irarA Kofio^ >Mr(ai) is thus explained by Produs {p, 198, 14—14). The first term describes the angles made by two straight lines whfXk one onlv is dividedf by the other, i.e. when one straight line meets another at a point which is not either of its extremi- ties, but is not itself produced beyond the point of meeting. When the first straight line is produced, so that the lines cross at the pomt, they make two pairs of vertuai angles (whidi are more clearly described as vertically opposite angles), and which are so called because their converrcnce is from opposite directions to one point (the intersection of the lines) as vertex («»pi^?). 36. at the point of section, literally "at the section," rpftt r% ro/ii. This theorem, according to Eudemus, was first discovered by Thales, but found its scientific demonstration in Euclid (Proclus, p. 299, 3—^). Proclus gives a converse theorem which may be stated thus. If a straight line is met at one and the same point intermediate in its length by two other straight lines on different sides of it and such as to mahe the vertical angles equals the latter straight lines are in a straight line with one another. The proof need not be given, since it is almost self-evident, whether (i) it is dirtet, by means of i. 13, 14, or (2) indirect, by reductio ad absurdum depending on I. 15. The balance of MS. authority seems to be against the genuineness of this' Porism, but Proclus and Psellus both have it. The word is not here used, as it is in the title of Euclid's lost Porisms, to si^ify a particular class of independent propositions which Proclus describes as bemgm some sort intermediate oetween theorems and problems (requiring us, not to bring a thing into existence, but to^iM/somethmg which we know to exist). Porism has here (and wherever the term is used in the Elements) its second meaning ; it is what we call a corollary, i.e. an incidental result springing from the proof of a theorem or the solution of a problem, a result not directly sought but appearing as it were by chance without any additional labour, and constituting;, as Produs sa3rs, a sort of windfall (^/acuov) and bonus (ic^os). These Ponsms appear in both the I. 15, 16] PROPOSITIONS IS, 16 279 geometrical and arithmetical Books of the Elements^ and may either result from theorems or problems. Here the Porism is geometrical, and springs out of a theorem ; vii. 2 affords an instance of an arithmetical Porism. As an instance of a Porism to a problem Proclus cites " that which is found in the second Book'' (ro iv rf Scvr^ fttpkuf ccifuvov) ; but as to this see notes on II. 4 and IV. 15. The present Porism, says Produs, formed the basis of ''that paradoxical theorem which proves that only the following three (regular) polygons can fill up the whole space surrounding one point, the equilateral triangle, the square, and the equilateral and equiangular hexagon." We can in fact place round a point in this manner six equilateral triangles, three regular hexagons, or four squares. "But only the angles of these regular figures, to the number specified, can make up four right angles : a theorem due to the Pythagoreans." Proclus further adds that it results from the Porism that, if any number of straight lines intersect one another at one point, the sum of all the angles so formed will still be equal to four right angles. This is of course what is . generally given in the text-books as Corollaiy 2. Proposition 16. In any triangle^ if one of the sides be produced^ the exterior angle is greaier than either of the interior and opposite angles. Let ABC be a triangle, and let one side of it BC be produced to D\ 5 I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA, BAC. Let A Che bisected at £ [i. lo], and let B£ be joined and produced 10 in a straight line to £] let £Fhe made equal to B£[i. 3], let FC be joined [Post, i], and let A C be drawn through to G [Post. 2]. Then, since A£ is equal to £Ct cs and B£ to £F, the two sides A£, £B are equal to the two sides C£, £F respectively ; and the angle A£B is equal to the angle F£Cy for they are vertical angles. [i. 15] JO Therefore the base AB is equal to the base FC, and the triangle AB£ is equal to the triangle CF£, and the remaining angles are equal to the remaining angles , respectively, namely those which the equal sides subtend ; [i. 4J therefore the angle BAB is equal to the angle ECF. a8o BOOK I [l i6 ; But the angle ECD is greater than the angle ECF\ therefore the angle ACD is greater than the angle BAE. Similarly also, if BC be bisected, the angle BCG, that is, the angle ACD [i. 15], can be proved greater than the angle ABC as well. Therefore etc. Q. E. d. 1. the exterior angle, literally ** the ontside angle,** ^ ifnh^ ymUa. 2. the interior and opposite angles, rOir irr^ jral dhrtroyrlar 7««Mi)ir. II. let AC be drawn through to O. The word is &4x^«#, a variatioo on the more usual iKPifiXili^ew, "let it ht producetL'* II. CFE, in the text •• FEC' As is well known, this proposition is not universally tnie under the Riemann hypothesis of a space endless in extent but not infinite in size. On this hypothesis a straight line is a *' closed series" and returns on itself; and two straight lines which have one point of intersection have another point of intersection also, which bisects the whole lengdi of the straight line measured from the first point on it to the same point again; thus the axiom of Euclidean geometry that two straight lines do not enclose a space does not hold. If 4A denotes the finite length of a straight line measured from any point once round to the same point again, a A is the distance between the two intersecticHis of two straight lines which meet Two points A^ B do not determine one sole straight line unless the distance between them is diffioent firom 2A. ^ In order that there may only be one perpendicular fix>m a point C to a straight line AB^ C must not be one of the two " poles " of the straight line. Now, in order that the proof of the present proposition may be universally valid, it is necessary that C/* should always fall within the angle ACD so that the angle ACFmay be less than the angle ACD. But this will not always be so on the Riemann hypothesis. For, (i) if BE is equal to A, so that BF is equal to 2 A, FmW be the second point in which BE and BD intersect ; i.e. /•will lie on CD, and the angle ACF will be egua/ to the angle ACD. In this case the exterior angle ACD will be e^al to the interior angle BAC> ^ (2) If BE is greater than A and less than 2A, so that BF is greater than 2A 1 and less than 4A, the angle ACF will be greater than the angle ACD, and | therefore the angle ACD will be less than the interior angle BAC. Thus, e.g.y | in the particular case of a right-angled triangle, the angles other than the right j angle may be (i) both acute, (2) one acute and one obtuse, or (3) both obtuse according as the perpendicular sides are (i) both less than A, (2) one less and the other greater than A, (3) both greater than A. Proclus tells us (p. 307, 1 — 12) that some combined this theorem with the next in one enunciation thus: In any triangle, if one side be produced, the exterior angle of the triangle is greater than either of the interior and opposite angles, and any two of the interior angles are less than two right angles, the combination having been suggested by the similar enunciation of EucUd l 32, In any trian^, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the trian^ are equal to two right angUs. The present proposition enables Proclus to prove what he did not succeed in estabhshing conclusively in his note on 1. 12/ namely ^BcaX from one point there cannot be drawn to the same straight line three straight lines equal in UngtK 1. i6, 17] PROPOSITIONS 16, 17 381 I let AB, AC, AD be all equal, £, C, D being in a equal, the angles For, if possible, straight line. Then, since AB, AC are ABC, ACB are equal. Similarly, since AB, AD are equal, the angles ABD, ADB are equal. Therefore the angle ACB is equal to the angle A DC, i.& the exterior angle to the interior and opposite angle: which is impossible. Proclus next (p. 308, 14 sqq.) undertakes to prove by means of i. 16 that, if a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle, the tufO straight lines will not form a triangle or meet, for in that case the same angle would be both greater and equal. The proof is really equivalent to that of EucL i. 27. V BE falls on the two straight lines AB, CD in such a wa^ that the angle CDE is equal to the interior and opposite angle ABD, AB and CD cannot form a triangle or meet. For, if they did, then (by i. 16) the angle CDE would be greater than the an^le ABD, while by the hypothesis it is at the same time equal to it Hence, says Proclus, in order that BA^ DC may form a triangle it is necessary for them to approach one another in the sense of being turned round one pair of corresponding extremities, e.g. B, D, so that the other extremities A, C come nearer. This may be brought about in one of three ways: (i) AB may remain fixed and CD be turned about D so that the angle CDE increases ; (2) CD may remain fixed and AB be turned about B so that the angle ABD becomes smaller; (3) both AB and CD may move so as to make die angle ABD smaller and the angle CDE larger at the same time. The reason, then, of the straight lines AB, CD coming to form a triangle or to meet is (says Proclus) the movement of the straight lines. Though he does not mention it here, Proclus does in another passage (p. 371, 2 — 10, quoted on p. 207 above) hint at the possibiUty that, while i. 16 may remain universally true, either of the straight ?ixit& BA, DC (or both together) may be turned through any angle not greater than a certain finite angle and yet may not meet (the Bolyai-Lobachewsky hypothesis). r Proposition 17. In any triangle two angles taken together in any manner -are less than two right angles. Let ABC be a triangle ; I say that two angles of the triangle ABC taken together in any manner are less than two right angles. For let BC be produced to I). [Post. 2] Then, since the angle A CD is an exterior angle of the triangle ABC, it is greater than the interior and opposite angle ABC [1. 16] 282 BOOK I [l 17 Let the angle ACB be added to each ; therefore the angles A CD, ACB are greater than the angles ABC, BCA. But the angles A CD, ^C^^raj equal to two right angles. [I. «3] Therefore the angles ^/^|^ BCA are less than two right imilarly we can prove thitoythe angles BAC, ACB are angles. f we can prove uiiMyuic au^ica ajxt,\^, xt,\^aj mc th an Ji^o right^anglc^and so are the angles CAB, AiclSm^.,^ pi %. Thereiore>6tc. ">'sk*.,.^ I ^^^^""^^^^ ^* ^' ^' I. taken Sogether in any manner, vd^rT^MraXaM/fftv^^MFai, Le. any pair added togetlier. V As in his note on the previous propodtiony Produs tries to state the comu of the property. He takes the case of two straight lines forming right anises with a transversal and observes that it is the convergence of the straight lifus towards one another (irvvcvo'if rwf cv^coSy), the lessening oi the two right angles, which produces the triangle. He will not have it that the &ct of the exterior angle being greater than the interior and opposite angle is the cause of the property, for the odd reason that '' it is not necessary that a side should be produced, or that there should be any exterior angle constructed... and how can what is not necessary be the cause of what is necessary?" (p. 311, 17 — 21). Agreeably to this view, Proclus then sets himself to prove the theorem without producing a side of the triangle. Let ABC be a triangle. Take any point D on BC, and join AD. Then the exterior angle ADC of the triangle ABD is greater than the interior and opposite angle ABD. Similarly the exterior angle ADB of the triangle ADC is greater than'tlie interior and opposite angle ACD. Therefore, by addition, the angles ADB, ADC are together greater than the angles ABC, ACB. But the angles ADB, ADC are equal to two right angles ; therefore the angles ABC, ACB are less than two right angles. - Lastly, Proclus proves (what is obvious from this proposition) that there cannot be more than one perpendicular to a straight line from a point without it. For, if this were possible, two of such perpendiculars would form a triangle in which two angles would be right angles: which is impossible, since any two angles of a triangle are together less tbuui two right angles. I. i8] PROPOSITIONS 17, 18 Proposition 18. 283 In atiy triangle the greater side subtends the greater angle. For let ABC be a triangle having the side AC greater than AB ; I say that the angle ABC is also greater than the angle BCA. For, since AC\^ greater than AB, let AD be made equal to AB [i. 3], and let BD be joined. Then, since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB. [l 16] But the angle ADB is equal to the angle ABD, since the side AB is equal to AD ; therefore the angle ABD is also greater than the angle ACB] therefore the angle ABC is much greater than the angle ACB. Therefore etc. Q. E. D. In the enunciation of this proposition we have ^orc/yecv (''subtend 'O used with the simple accusative instead of the more usual inrh with accusative. The latter construction is used in the enunciation of i. 19, which otherwise only differs from that of 1. 18 in the order of the words. The point to remember in order to distinguish the two is that the daium comes first and the quauitum second, the datum being in this proposition the greater side and in the next the greater angie. Thus the enunciations are (l. 18) wwnht rpiyiiifov if /i/d{taif wXtvpik rifif fktl^wa yw^lnM intvrthti and (l. 19) waanht rpvyfiifw ^6 rV jul^a ytfifUuf ^ tMtl{m irXcvpd ^or€h€t. In order to keep the proper order in En^lidi we must use the passive of the verb in i. 19. Aristotle quotes the result of i. 19, using the exact wording, ord Tdp "Hiw fuli^ ytfiflaw inrorc^ci (Afeteorologica III. 5, 376 a is). ''In order to assist the student in remembering which of these two propositions [i. 18, 19] is demonstrated directly and '" which indirectly, it may be observed that the order is similar to that in i. 5 and i. 6" (Todhunter). An alternative proof of i. 18 given by Porphjny (see Proclus, pp. 315, 11 — 316, 13) is interesting. It starts by supposing a length equal to A£ cut off from the other end of -^C; that is, CD and not AD is made equal to AB. Produce AB to E so that BE is equal to AD, and join EC Then, since AB is equal to CD, and BE to AD, AE is equal to AC 384 BOOK I [l 18, 19 Therefore the angle ARC is equal to the angle ACE. Now the angle ABC is greater than the angte AEC^ [l 16] and therefore greater than the angle ACE. Hence, a fortiori^ the angle ABC is greater than the angle ACB. Proposition 19. In any triangle the greater angle is subtended by the greater side. Let ABC be a triangle having the angle ABC greater than the angle BCA ; I say that the side AC is also greater than the side AB. For, if not, -^C is either eaual to AB or less. Now -^C is not equal to AB ; for then the anc^le ABC would also have been equal to the angle ACB ; [i. sj but it is not ; therefore AC is not equal to AB. Neither is -^C less than AB, for then the angle ABC would also have been less than the Single ACB; [1. iS] but it is not ; therefore AC is not less than AB. And it was proved that it is not equal either. Therefore -^C is greater than AB. Therefore etc. Q. e. d. This proposition, like i. 6, can be proved by merely /dguai deduction from I. 5 and I. 18 taken together, as pointed out by De Morgan. The seneral form of the argument u^ by De Morgan is given in his Format Zcgic {iS4j\ p. 25, thus : ^^ Hypothesis. Let there be any number of propositions or assertions — three for instance, X, Y and Z— of which it is tiie property that one or the other must be true, and one only. Let there be three other propositions Py Q and ^ of which it is also the property that one, and one only, must be true. Let it be a connexion of those assertions that : when X is true, P is true, when yis true, Q is true, when Z is true, ^ is true. Consequence: then it follows that, when P)S true, X is true, •when Q is true, K is true, when R is true, Zis true." I I [1 5] [I- i8] [' .6] [I- ^9] I. 19] PROPOSITIONS 18, 19 385 To apply this to the case before us, let us denote the sides of the triangle ABC by tf, by r, and the angles opposite to these sides by A^ B^ C respectively, and suppose that a is the base. Then we have the three propositions, when b is equal to r, ^ is equal to C, when b is greater than r, B is greater than C, 1 when b is less than r, B is less than C, / and it follows lagUally that, when B is equal to C, b is equal to r, \ when B is greater than C, ^ is greater than r, 1 when B is less than C, ^ is less than e. / Reductio ad absurdum by exhaustion.- Here, says Proclus (p. 318, 16 — 23), Euclid proves the impossibili^ "by means of division^* {U huupkr^ioi). This means simply the separation of different hypotheses, each of which is inconsistent with the truth of the theorem to be provcKl, and which therefore must be successively shown to be impossible. If a straight line is not greater than a straight line, it must be either equal to it or less ; thus in a reductio ad absurdum intended to prove such a theorem as i. 19 it is necessary to dispose successively of Hue hypotheses inconsistent with the truth of the theorem. Alternative (direct) proof. Proclus gives a direct proof (pp. 319 — 321) which an-Nairia also has and attributes to Heron. It requires a lemma and is consequently open to the slight objection of separating a theorem from its converse. But the lemma and proof are worth giving. Lemma. Jfan angle of a triangle be bisected and the straight line bisecting it meet the base and dMde it into unequal parts^ the sides containing the angle will be uncqucUy and the greater will be that which meets the greater segment of the base^ and the las thcU which meets the less. Let ADy the bisector of the angle A of the triangle ABC^ meet BC in Z>, making CD greater than BD, I say that AC is greater than AB. Produce AD ioE so that DE is equal to AD. And, since Z>C is greater than BD^ cut oflDFeqvaltoBD. Join EFand produce it to G. Then, since the two sides AD, DB are equal to the two sides ED^ DF^ and the vertical angles at D are equal, AB is equal to EF^ and the angle DEF io the angle BAD, i.e. to the angle DAG (by hypothesis). Therefore AG\s equal to EG^ [1. 6] and therefore greater than EF, or AB. Hence, afartiori, AC\& greater than AB. U 286 BOOK I [i. 19, to Proof of I. zg. Let ABC be a triangle in which the angle ABC is greater than the ang^ ACB* Bisect BC at Z>, join AD^ and produce it to j? so that DE is equal to AD, ]o\ti BE. Then the two sides BD^ DE are equal to the two sides CD^ DA^ and the vertical angles at Z> are equal ; therefore BE is equal to AC^ and the angle DBE to the angle at C But the angle at C is less than the angle ABC\ therefore the angle DBE is less than the angle ABD. Hence, if BF bisect the angle ABE^ BF meets AE between A and D. Therefore EF is greater than FA. It follows, by the lemma, that BE is greater than BA, that is, AC is greater than AB. Proposition 20. In any triangle two sides taken together in any manner are greater than the remaining one. For let ABC be a triangle ; I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely BAy AC greater than BC, AB, BC greater than AC, BCy CA greater than AB. For let BA be drawn through to the point /?, let DA be made equal to CA, and let DC be joined. Then, since DA is equal to AC, the angle ADC is also equal to the angle ACD; [I.S] therefore the angle BCD is greater than the angle ADC. [C N. 5] And, since DCB is a triangle having the angle BCD greater than the angle BDC, and the greater angle is subtended by the greater side, [i. 19] therefore DB is greater than BC. I. 20] PROPOSITIONS 19, ao 287 But DA is equal to AC; therefore BA, AC are greater than BC. Similarly we can prove that AB, BC are also greater than CA, and BC, CA than AB. Therefore etc. Q. E, D. It was the habit of the Epicureans, says Proclus (p. 323), to ridicule this theorem as being evident even to an ass and requiring no proof, and their all^ation that the theorem was ''known" (yvwpifiov) even to an ass was based on the fact that, if fodder is placed at one angular point and the ass at another, he does not, in order to get to his food, traverse the two sides of the triangle but only the one side separating them (an argument which makes Savile exclaim that its authors were "digni ipsi, qui cum Asino foenum essent," p. 78). Proclus replies truly that a mere perception of the truth of the theorem is a different thing from a scientific proof of it and a knowledge of the reason why it is true. Moreover, as Simson says, the number of axioms should not be increased without necessity. Alternative Proofs. Heron and Porphyry, we are told (Proclus, pp. 323 — 6), proved this tfieorem in different ways as follows, without producing one of the sides. J^rs/ proof. Let ABC be the triangle, and let it be required to prove that the sides BA, AC are greater than BC Bisect the angle BA C by AD meeting BC in D. Then, in the triangle ABD, . the exterior angle ADC is greater than the interior and opposite angle BAD, [i. 16] that is, greater than the angle DAC Therefore the side ^C is greater than the side CD. [i. 19] Similarly we can prove that AB is greater than BD, Hence, by addition, BA, ACsoregjreaittr than BC. Second proof . This, like the first proof, is direct. There are several cases to be considered. ii) If the triangle is equilateral, the truth of the proposition is obvious. 2) If the triangle is isosceles, the proposition needs no proof in the case (a) where each of the equal sides is ^eater than the base. (b) If the base is greater than either of the other sides, we have to prove that the sum of the two equal sides is greater than the base. Let BC be the base in such a triangle. Cut off from BC a length BD equal to AB, and join AD. Then, in the triangle ADB, the exterior angle ADC is greater than the interior and opposite angle BAD. • [i. 16I Similarly, in the triangle ADC, the exterior angle ADB is greater than the interior and opposite angle CAD. 388 BOOK I [l 3o By addition, the two angles BDA^ ADC are together greater than the two angles BAD, DAC (or the whole angle BAC). Subtracting the equal angles BDA, BAD^ we have tfie angle ADC greater than the angle CAD. It follows that ^C is greater than CD\ [i. 19] and, adding the equals AB^ BD respectively, we have BA^ AC together greater than BC (3) If the triangle be scakm^ we can arrange the sides in order of lensth. Suppose BC is the greatest, AB the intermediate and AC the least side. Then it is obvious that AB, BC are together greater than AC^ and BC^ CA together greater than AB, It only remains therefore to prove that CA, AB are together greater than BC. We cut off from BC a length BD equal to the adjacent side^ join AD^ and proceed exactly as in the above case of the isosceles trian^e. Thirdpraof. This proof is by rtductio ad absurdum. Suppose that BC is the greatest side and, as before, we have to prove that BA, ACsLre greater than BC. If they are not, they must be either equal to or less than BC (i) Suppose BA, AC are together equal to BC. From BC cut off BD equal to BA, and join AD. It follows from the hypothesis that DC is equal to AC. Then, since BA is equal to BD, the angle BDA is equal to the angle BAD. Similarly, since ^C is equal to CD, the angle CDA is equal to the angle CAD. By addition, the angles BDA, ADC are together equal to the whole angle BAC. That is, the angle BAC is equal to two right angles : which is impossible. (2) Suppose BA, AC are together less than BC. From BC cut off BD equal to BA, and from CB cut off CE equal to CA. Join AD, AB. In this case, we prove in the same way that the angle BDA is equal to the angle BAD, and the angle CEA to the angle CAE. By addition, the sum of the angles BDA, AEC is equal to the sum of the angles BAD, CAE. Now, by I. 16, the angle BDA is greater than the angle DAC, and theiefore, a fariiari, greater than the angle EAC. Similarly the angle AEC is greater than the angle BAD. S Hence the sum of the angles BDA, AEC is greater than the sum of the ' angles BAD, EAC. But the former sum was also equal to the lAtter : which is impossible. ! I. ai] PROPOSITIONS 20, 21 389 Proposition 21. If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a grecUer I angle. j On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be con- I structed meeting within the triangle ; I I say that BD, DC are less than the remaining^ two sides MO of the triangle BA, AC, but contain an angle BvC greater I than the angle BAC For let BD be drawn through to E. Then, since in any triangle two , sides are greater than the remaining ii5 one, [i. 20] therefore, in the triangle ABE, the \ two sides -^jff, AE are greater than BE. Let EC be added to each ; therefore BA, AC are greater than BE, EC |ao Again, since, in the triangle CED, \ the two sides CE^ ED are greater than CD, \ let DB be added to each ; therefore CE, EB are greater than CD, DB. But BA^ AC were proved greater than BE, EC; therefore BA, AC are much greater than BD, DC. Agrain, since in any triangle the exterior angle is greater than the interior and opposite angle, [i. 16] therefore, in the triangle CDE, the exterior angle BDC is greater than the angle CED. o For the same reason, moreover, in the triangle ABE also, the exterior angle CEB is greater than the angle BAC. But the angle BDCvias proved greater than the angle CEB; therefore the angle BDC is much greater than the angle BAC. »^ Therefore etc. q. e. d. 3. be constructed... meeting within the triangle. The word "meeting" is not in the Greek, where the words are irrhs cveraBCMtv, ffwlffroffSai is the word lued of con- structing two straight lines to a point (cf. i. 7) or^ as to form a triangle ; but it is necessaiy in Engksh to indicate that they nuet. 3. the straight lines so constructed. Observe the elegant brevity of the Greek ol H. B. 19 «90 BOOK I [i. ti 1 The editors generally call attention to the fiiurt that the lines dnwn within the triangle in this proposition must be drawn, as the enunciation says, from the ends of the side ; otherwise it is not necessary that their sum should be less than that of the remaining sides of the triangle. Proclus (p. 337, 1 2 sqq.) gives a simple illustration. Let i^^C be a right-angled triangle. Take any point D on BC^ join DA^ and cut off from it DE equal to AB, Bisect AE at A and join FC. Then shall CF, FD be together greater than CA^ AB. For CF, FE are equal to CF, FA, and therefore greater than CA. Add the equals ED, AB respectively ; therefore CF, FD are together greater than CA, AB. Pappus gives the same proposition as that just proved, but follows it up by a number of others more elaborate in character, selected apparently from " the socalled paradoxes " of one Erycinus (Pappus, in. p. 106 sqq.). Thus he proves the following : 1. In any triangle, except an equilateral triangle or an isosceles triangle with base less than one of the other sido, it is possible to construct on the base and within the triangle two straight lines the sum of which is equal to the sum of the other two sides of the triangle. 2. In any triangle in which it is possible to construct two straight lines on the base which are equal to the sum of the other two sides of the triangle it is also possible to construct two others the sum of which isgnaierXbBn that sum. 3. Under the same conditions, if the base is greater than either of the other two sides, two straight lines can be constructed in the manner described which are respectively greater than the other two sides of the triangle ; and the lines may be constructed so as to be respectively equal to the two sides, if one of those two sides is less than the other and each of them less than the base. 4. The lines may be so constructed that their sum wiU bear to the sum of the two sides of the triangle any ratio less than 2:1. As a specimen of the proofs we will give that of the proposition whidi has been numbered (i) for the case where the triangle is isosceles (Pappus, iii. pp. 108 — no). I. 2i] PROPOSITION 21 391 Let ABC he an isosceles triangle in which the base ^C is greater than either of the equal sides AB, BC. With centre A and radius AB describe a circle meeting ACin D. Draw any radius AEJ^such that it meets ^Cin a point jF outside the circle. Take any point G on EFy and through it draw (7^ parallel to AC. Take any point JTon GJfy and draw J^L parallel to IrA meetmg i^C in Z. From BC cut off -5-A^ equal to EG. Thus ^^, or LJiiy is equal to the sum of AB^ BN^ and CTV^is less than LK, Now GFy FHzxe together greater than GH, and CH, /T^ together greater than CK. Therefore, by addition, CA FGy BK9je together greater than CK, HG. Subtracting HK from each side, we see that CF, FG are together greater than CK, KG \ therefore, if we add AG to each, AF, FC are together greater than AG, GK, KC And AB, BCsLTt together greater than AF, FC. [i. 21] Therefore AB, BCaie together greater than AG, GK, KC But, by construction, AB, BN dse together equal to ^^; therefore, by subtraction, NC is greater than GK, KC, and a fortiori greater than KC. Take on KC produced a point Jlfsuch that KM is equal to NC; with centre A!* and radius JTJl/' describe a circle meeting CL in O, and join KO. Then shall LK, KO be equal to AB, BC. For, by construction, ZK is equal to the sum of AB, BN, and KO is equal to NC\ therefore LK, KO sure together equal to AB, BC It is after i. 31 that (as remarked by De Morgan) the important proposition about the perpendicular and obliques drawn from a point to a straight line of unlimited length is best introduced : 0/ all straight lines that can be drawn to a given straight line of unlimited length from a given point without it: (a) the perpendicular is the shortest; (d) of the obliques, that is the greater the foot of which is further from the perpendicular ; {c) given one oblique, only one other can be found of the same length, namely that the foot of which is equally distant with the foot of the given one from the perpendicular, but on the other side of it. Let A be the given point, BC the given straight line; let AD be the perpendicular from A on BC, and AE, AF any two obliques of which AF makes the greater angle inihAD. Produce AD to A', making A'D equal to AD, and join A'E, A F. Then the triangles ADE, ADE are equal in all respects ; and so are the triangles ADF, A'DF Now (i) in the triangle AEA' the two sides AE, EA are greater than AA [i. 20J, that is, twice AE is greater ^lan twice AD. 19 — 2 392 BOOK I [mi, 99 Therefore A£ is greater than AD. (2) Since AE, AE are drawn to £, a point within the triangle AFA\ AF, FA are together greater than AE^ EA\ [l 21] or twice AFis greater than twice AE. Therefore ^^is greater than AE. (3) Along DB measure off DG equal to DF^ and join AG. The triangles AGD, AFD are then equal in all respects, so that the angles GAD^ FAD are equal, and AG is equal to AP. Proposition 22. Out of three straight lines, which are equal to three given straight lines, to construct a triangle : thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. [i. 20] Let the three given straight lines be A, B, C, and of these let two taken together in any manner be greater than the remaining one, namely A, B greater than C, A, C greater than B, and B, C greater than A ; thus it is required to construct a triangle out of straight lines equal to A, B, C. A- B- C- Let there be set out a straight line DE, terminated at D but of infinite length in the direction of E, and let DF be made equal to A, FG equal to B, and GH equal to C. [i- 3] With centre F and distance FD let the circle DKL be described ; again, with centre G and distance GH let the circle KLH be * described ; and let KF, KG be joined ; I say that the triangle KFG has been constructed out of three straight lines equal to A, B, C. i I. at] PROPOSITIONS ai, 22 293 ( ( For, since the point /^ is the centre of the circle DKL, FD is equal to FK. But FD is equal to A ; therefore KF is also equal to A. Again, since the point G is the centre of the circle LKH^ GH is equal to GK. But GH is equal to C \ therefore KG is also equal to C And FG is also equal to B ; therefore the three straight lines KF, FG, GK are equal to the three straight lines A, B, C. Therefore out of the three straight lines KF, FG, GK, which are equal to the three given straight lines A, B, C, the triangle KFG has been constructed. ^ Q. E. F. 3 — 4. This is the first case in the Eltments of a iiapifffjM to a problem in the sense of a statement of the conditions or limits of the possibility of a solution. The criterion is of coarse supplied by the preceding proposition. 3. thus it is necessary, lliis is usaally translated (e.g. by Williamson and Simson) **£ui it is necessary,'' which is however inaccurate, since the Greek is not M 84 but 8«t d^. The words are the same as those used to introduce the tiopw^Lbi in the other sense of the "definition" or "particular statement** of a construction to be effected. Hence, as in the latter case we say "thus it is required *' (e.g. to bisect the finite straight line AB, i. lo), we should here translate *' thus it is necessary.^ 4. To this enunciation all the mss. and Boethius add, after the iiOfMiiM^ the words "because in any triangle two sides taken together in any manner are greater than the remaining one.** But ub explanation has the appearance of a gloss, and it is omitted by Produs and Campanus. Moreover there is no corresponding addition to the iwpLrikbt of VI. «8. It was early observed that Euclid assumes, without giving any reason, that the circles drawn as described will meet if the condition that any two of the straight lines A, B, C are together greater than the third be fulfilled. Proclus (p- 33I9 S sqq.) argues the matter by means of reductio ad absurdum, but does not exhaust the possible hypotheses inconsistent with the contention. He says the circles must do one of three things, (i) cut one another, (2) touch one another, (3) stand apart (Sicoravat) from one another. He then considers the hypotheses {a) of their touching externally, (p) of their being separated from one another by a space. He should have considered also the hypothesis {f) of one circle touching the other internally or lying entirely within the other without touching. These three hypotheses being successively disproved, it follows that the circles must meet (this is the line taken by Camerer and Todhunter). Simson says: "Some authors blame Euclid because he does not demonstrate that the two circles made use of in the construction of this problem must cut one another : but this is very plain from the determination he has given, namely^ that any two of the straight lines DF, FG, GHmyi%\. be greater than the third. For who is so dvdl, though only beginning to learn the Elements, as not to perceive that the circle described from the centre F, at the distance FD, must meet FH betwixt F and H, because FD 18 less than FH\ and that, for the like reason, the drcle described from the 394 BOOK I [i. 33, 33 centre G at the distance GH must meet DG betwixt D and G ; and that these circles must meet one another, because FD and GH are together greater than FG:' We' have in fact only to satisfy ourselves that one of the drcles, e.g. that with centre G^ has at least one point of its circumference outside the other circle and also at least one point of its circumference inside the same circle ; and this is best shown with reference to the points in which the first circle cuts the straight line BE. For ^i) FH^ being equal to the sum of B and C, is greater than A^ i.e. than the radius of the circle with centre F^ and therefore H is outside that circle. (2) If GM be measured along (7^ equal to GH or C, then, since GM is either (a) less or greater than GF, JVwill fidl either {a) between G and ^or {b) beyond /^towards Z>; in the first case {a) the sum of FM and C is equal to FG and therefore less than the sum of A and C, so that FM is less than A or FD; in the second case {i) the sum of MF and FG^ i.e. the sum of MF and By is equal to GMix C «m1 therefore less than the sum of A^ and B^ so that MF is less than A ^ FD\ hence in either case M &Us within the circle with centre F. It being now proved that the circumference of the circle with centre G has at least one point outside, and at least one point inside, the drde with centre F^ we have only to invoke the Principle of Continm'ty, as we have to do in I. I (cf. the note on that proposition, p. 242, where the necessary postulate is stated in the form suggested by Killing). That the construction of the proposition gives only two points of intersection between the circles, and therefore only two triangles satisfying the condition, one on each side of FG^ is dear firom i. 7, which, as bdbre pointed out, takes the place, in Book i., of iil 10 proving diat two ciides cannot intersect in more points than two. Proposition 23. On a given straight line and at a point an it to construct a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, A the point on it, and the angle DCE the given rectilineal angle; # thus it is required to construct on tne given straight line AB, and at the point A on it, a rectilineal angle equsu to the given rectilineal angle DCE. On the straight lines CD, CE respectively let the points /?,.£* be taken at random ; let DE be joined, arid out of three straight lines which are equal to the three I. 23] PROPOSITIONS 22, 23 29s Straight lines CD, DE, CE let the triangle AFG be con- structed in such a way that CD is equal to AF, CE to AG, and further DE to FG. ^i. 22] Then, since the two sides DC, CE are equal to the two ^ sides FA, AG respectively, ' and the base DE is equal to the base FG, the angle DCE is equal to the angle FAG. [i. 8] \ Therefore on the given straight line AB, and at the point I -^ on it, the rectilineal angle FAG has been constructed equal f to the given rectilineal angle DCE. This problem was, according to Eudemus (see Proclus, p. 333, 5), "rather the discovery of Oenopides," from which we must apparently infer, not that Oenopides was the first to find any solution of it, but that it was he who dis- covered the particular solution given by Euclid. (Cf. Bretschneider, p. 65.) The editors do not seem to have noticed the fact that the construction of the triangle assumed in this proposition is not exactly the construction given in I. 22. We have here to construct a triangle on a certain finite straight line AG9& base; in i. 22 we have only to construct a triangle with sides of given length without any restriction as to how it is to be placed. Thus in i. 22 we set out any line whatever and measure successively three lengths along it b^inning from the given extremity, and what we must r^ard as the base is the intermecUate length, not the length banning at the given extremity of the straight line arbitrarily set out Here the base is a given straight line abutting at a given point Thus the construction has to be modified somewhat from I ITb that of the preceding proposition. We must measure AG along AB so that AG\& equal to CE (or CD), and C^ along GB equal to DE\ and then we must produce BA, in the opposite direction, to ^ so that AF\& equal to CD (or CE, ii AG has been made equal to CD). Then, by drawing circles (i) with centre A and radius AF, (2) with centre G and radius GIf, we determine J^, one of their points of intersection, and we prove that the triangle J^AG is equal in all respects to the triangle DCE, and then that the angle at A is equal to the angle DCE. I think that Proclus must (though he does not say so) have felt the same difficulty with regard to the use in i. 23 of the result of i. 22, and that this is probably the reason why he gives over again the construction which I have given above, with the remark (p. 334, 6) that "you may obtain the construction of the triangle in a more instructive manner (StSaoncaXaci^cpov) as follows." Proclus objects to the procedure of Apollonius in constructing an angle under the same conditions, and certainly, if he quotes Apollonius correctly, &e latter's exposition must have been somewhat slipshod. S96 BOOK I [i. S3» 94 "He takes an angle CDE at random," says Produs (p. 335, 19 sqq.X ''and a straight line AB^ and with centre D and distance CD describes the circumference C£, and in die same way with centre A and distance AB the circumference FB. Then, cutting off FB equal to CE, he joins AF. And he declares that the aisles A^ D standii^ on equal circumferences are equaL" _ In the first place, as Proclus remaiks, it should be E premised that AB is equal to CD in order that the ^^^ circles may be equal; and the use of Book in. for such an elementary construction is objectionable. The omission to state that AB must be taken equal to CD was no doubt a slip, if it occurred. And, as r^ards the equal angles '' standing on equal drcum- . ^ .ferences," it would seem possible mat Apollonius said this in explanation^ for the siJce of brevity, rather than by way of proof. It seems to me probable that his construction was only given from the point of view olfrachcal^ not theoretical, geometry. It really comes to the same thing as Euclid's except that DC is taken equal to DE. For cutting off the arc BF equal to the arc CE can only be meant in the sense of measuring the M^n/ C£, say, with a pair of compasses, and dien drawing a circle with centre B and radius eqiial to the chord CE. Apollonius' direction was therefore probably intended as a practical short cut, avoiding the actual drawing of the chords CE^ BF^ which, as well as a proof of the ec^uali^ in all respects of the triangles CDE^ BAF^ would be required to estabhsh tluonikaUy the correct- ness of the construction. Proposition 24. If two triangles have the two sides equal to two sides respectively^ but have the one of the angles contained by the equal straight lines greater than the other, they will also have the bc^e greater than the base. S Let ABC, DEF be two triangles having the two sides AB^ -^C equal to the two sides DE, Z?/^ respectively, namely AB to DE, and -^C to DF, and let the angle at -^ be greater than the angle at D ; I say that the base BC is also greater than the base EF. 10 For, since the angle BAC is greater than the angle EDF, let there be constructed, on the straight line DE, and at the point D on it, the angle EDG IS equal to the angle BAC; [i. 23] let DG be made equal to either of the two straight lines AC, DF, and let EG, FG be joined. I I ' 1.24] PROPOSITIONS 23,24 297 i Then, since AB is equal to DE, and AC to DG, j ao the two sides BA, AC are equal to the two sides ED, DG, respectively ; . and the angle BAC is equal to the angle EDG ; therefore the base BC is equal to the base EG. [1. 4] V Again, since DF is equal to DG, * 25 the angle DGF is also equal to the angle DFG ; [i. 5] therefore the angle DFG is greater than the angle EGF. It Therefore the angle EFG is much greater than the angle * EGF. And, since EFG is a triangle having the angle EFG 30 greater than the angle EGF, and the greater angle is subtended by the greater side, [I. 19] the side EG is also greater than EF. But EG is equal to BC. Therefore BC is also greater than EF. f 35 Therefore etc. Q. E. D. 10. I have naturally left out the well-known words added by Simson in order to avoid the necessity of considering three cases : " Of the two sides DE, DF let DE be the side which is not greater than the other." I doubt whether Euclid could have been induced to insert the words himself, even if it had been represented to him that their omission meant leaving two possible cases out of consideration. His habit and that of the great Greek geometers was, not to set out all possible cases, but to give as a rule one case, generally the most difficult, as here, and to leave the others to the reader to work out for himself. We have already seen one instance in i. 7. Proclus of course gives the other two cases which arise if we do not first provide that DE is not greater than DF. (i) In the first case G may fall on EF produced, and it is then obvious that EO is greater than EF, (2) In the second case EG may fall below EF. If so, by 1. 21, DF, FE are together less than DG, GE. But DF is equal to DG \ there- fore EF is less than EG, i.e. than BC. These two cases are therefore decidedly simpler than the case taken by Euclid as typical, and could well be left to the ingenuity of the learner. If however after all we prefer to insert Simson's words and avoid the latter 2gB BOOK I [1.24 two cases, the proof is not complete unless we show that, with his assumptioo, /^must, in the figure of the proposition, &11 Mn» EG. De Morgan would make the following proposition precede: Every siraigki line drawn from the vertex of a trmngk to the base is less than the greaier of ike two sides, or than either if they are equal, and he would prove it by means of the proposition relating to perpendicular and obliques mven above, p. 391. But it is easy to prove directly that F &lls below EG, if DE is not greater than DG^ by the method employed by Pfleiderer, Lardner, and Todhunter. Let DF, produced if necessary, meet EG in H. Then the angle DHG is greater than the angle DEG\ [I. 16] and the angle DEG is not less than the angle DGE : [.. 18] therefore the angle DHG is greater than the angle DGH. Hence DH\% less than DG^ [i. 19] and therefore DH is less than DF. Alternative proof. Lastly, the modem alternative proof is worth giving. A D A Let Z>^ bisect die angle FDG (after the triangle DEG has been made equal in all respects to the triangle ABC^ as in the proposition), and let DH meet EG in H Join HF. Then, in the triangles FDH, GDH, the two sides FD, DHzx^ equal to the two sides GD, DH, and the included angles FDH, GDHzxt, eqiial ; therefore the base HF is equal to the base HG. Accordingly EG is equal to the sum of EH, HF\ and EH, HF are together greater than EF\ [i. 20] therefore EG, or BC, is greater than EF. Proclus (p. 339, 1 1 sqq.) answers by anticipation the possible question that might occur to any one on this propc^ition, viz. why does Euclid not compare the areas of the triangles as he does in i. 4 ? He observes that inequali^ of the areas does not follow from the inequality of the angles contained by the equal sides, and that Euclid leaves out all reference to the question boUi for this reason and because the areas cannot be compared without the help of the theory of parallels. *' But if^" says Proclus, *' we must anticipate what is to come and make our comparison of the areas at once, we assert that (i) if the an^ Af D — stiptposing that our argument proceeds with reference to the figure in the proposition — are {together) equeU to two right angles, the triangles I. 24, 25] PROPOSITIONS 24, 25 299 are proved equals (2) if greater than two right angles^ that triangie which has the greater angle is lesSy and (3) if they are less, greater" Proclus then gives the proof, but without any reference to the source from which he quoted the proposition. Now an-Nairiz! adds a similar proposition to i. 38, but definitely attributes it to Heron. I shall accordingly give it in the place where Heron put it Proposition 25. // two triangles have the two sides equal to two sides respectively, but have the base greater than the base, thiy will also have the one of the angles contained by the equal straight lines greater than the other. Let ABC, DEF be two triangles having the two sides AB^ AC equal to the two sides DE^ Z?/^ respectively, namely AB to DE, and -^C to DF\ and let the base BC be greater than the base EF\ I say that the angle BAC is also greater than the angle EDF. For, if not, it is either equal to it or less. • Now the angle BAC is not equal to the angle EDF\ I for then the base BC would also have been equal to the base , EF, [1.4] but it is not ; therefore the angle BAC is not equal to the angle EDF. Neither again is the angle BAC less than the angle EDF; for then the base BC would also have been less than the base EF, ^ [1.24] but it is not ; therefore the angle BAC is not less than the angle EDF. But it was proved that it is not equal either ; therefore the angle BAC is greater than the angle EDF. Therefore etc. Q. E. D. 300 BOOK I [1.95 De Morgan points out that ibis jvoposition (as also i. 8) is a purdy iogieai consequence of i. 4 and i. 34 in the same way as i. 19 and l 6 are purdy icgua/ consec^uences of i. 18 and 1. 5. If n, ^, c denote the sidesi A^ B, Cytid , angles opposite to them in a triangle ABQ and a^^ V^ /, A\ B, Ciht sides* and opposite angles respectivdy in a triangle A'BC^ i. 4 and i. 24 tdl us that, b^ c being respectively equal to V^ i^ (i) if ^ is equal to A\ then a is equal to a\ (2) if ^ is less than A\ then a is less than a', (3) if ^ is greater than A\ then a is greater than a' ; and it follows iogually that, (i) if a is equal to a', the angle A is equal to the angle A\ \u 8] (3) if a is less than a', ^ is less than A\ \ . . (3) if a is greater than «*, -^ is greater than A\ ] L'' *5j Two alternative proofs of this theorem are given by Produs (pp. 345 — 7), and they are both interesting. Moreover both are Hred. I. Proof by Menelaus of Alexandria. Let ABC, jDEFbe two triangles having the two sides BA, AC equal to the two sides EjD, DF, but the base ^C greater than the base EF. Then shall the angle at ^ be greater than the angle at Z>. From BC cut off BG equal to EF. At B, on the straight line BC, make the angle GBH {on the side of BG remote from A) equal to the angle FED. Midce BH equal to DE\ join HG, and produce it to meet AC in K. Join AH. Then, since the two sides GB, BH are equal to the two sides FE^ ED respectively, and the angles contained by them are equal, HG is equal to DFox AC^ \ and the angle BHG is equal to the angle EDF. ^ Now HK is greater than HG 01 AC, \ and a fortiori greater than AK\ therefore the angle KAH\& greater than the angle KHA. And« since AB is equal to BH, the angle BAH\% equal to the angle BHA. Therefore, by addition, the whole angle BA C is greater than the whole angle BHG, that is, greater than the angle EDF. 1. 25, a6] PROPOSITIONS 25, 26 301 II. Heron's proof. Let the triangles be given as before. Since BC is greater than EF^ produce EF to (r so that EG is equal to BC. Produce ED to If so that DIf is equal to I?F The circle with centre D and radius Z>^will then pass through If. Let it be described^ as FXIf. Now, since B:,!^ AC axe together greater than BQ and BA, AC Bie equal to ED^ />^ respectively, while BC is eqiial to EG^ EH is greater than EG. Therefore the circle with centre j^and radius EG will cut Elf^ and therefore will cut the circle already drawn. Let it cut that circle in JT, and join DK, KE. ^ Then, since Z> is the centre of the circle FKH^ iDK is eqiial to DF or AC. Similarly, since E is the centre of the circle KG^ EK\& equal to EG or BC^ And DE is equal to AB. Therefore the two sides BA^ AC 9st eqiial to the two sides ED^ DK respectively ; and the base BC is equal to the base EK\ therefore the angle BAC is equal to the angle EDK. Therefore the angle BAC is greater than the angle EDF. Proposition 26. I If two triangles have the two angles equal to two angles I respectively y and one side equal to one side^ namely, either the side adjoining the equal angles, or thai subtending one of the equal angles, they will also have the remaining sides equal to 5 the remaining sides and the remaining angle to the remaining angle. 302 BOOK I [l s6 Let ABC, DEF be two triangles having the two angles ABC, BCA equal to the two angles DEF, EFD respectivelv, namely the angle ABC to the angle DEF, and the ansle ^^BCA to the angle EFD; and let them also have one side equal to one side, first that adjoining the equal angles, namely BC to EF\ I say that they will also have the remaininc; sides equal to the remaining sides respectively, namely aS to DE and 15-^C to DF, and the remaining angle to the remaining angle, namely the angle BAC to the angle EDF. B For, if AB is unequal to DE, one of them is greater. Let AB be greater, and let BG be made equal to DE ; and let GC be joined. ao Then, since BG is equal to DE, and BC to EF, the two sides GB, BC are equal to the two sides DE, EF respectively; and the angle GBC is equal to the angle DEF; therefore the base GC is equal to the base DF, 25 and the triangle GBC is equal to the triangle DEF, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend ; [i. 4] therefore the angle GCB is equal to the angle DFE. But the angle DFE is by hypothesis equal to the angle BCA ; 30 therefore the angle BCG is equal to the angle BCA, the less to the greater : which is impossible. Therefore AB is not unequal to DE, and is therefore equal to it. But BC is also equal to EF-, 35 therefore the two sides AB, BC are equal to the two sides DE, EF respectively, and the angle ABC is equal to the angle DEF\ therefore the base ACv& equal to the base /?/% and the remaining angle BAC is equal to the remaining 40 angle EDF. [l 4] T 26] PROPOSITION 36 303 Again, let sides subtending equal angles be equal, as AB , to DE\ ' I say again that the remaining sides will be equal to the remaining sides, namely AC to DF and BC to EF, and ^45 further the remaining angle BAC is equal to the remaining angle EDF. 1 For, if BC is unequal to EF^ one of them is greater. Let BC be greater, if possible, and let BH be made equal to EF\ let Am be joined. 50 Then, since Bm is equal to EF, and AB to DE, the two sides AB, BH are equal to the two sides DE, EF respectively, and they contain equal angles ; therefore the base AH is equal to the base DF, and the triangle ABH is equal to the triangle DEF, 55 and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend ; [i. 4] therefore the angle BHA is equal to the angle EFD. But the angle EFD is equal to the angle BCA ; therefore, in the triangle AHC, the exterior angle BHA is 60 equal to the interior and opposite angle BCA : which is impossible. [i. 16] Therefore BC is not unequal to EF, and is therefore equal to it. But AB is also equal to DE ; 65 therefore the two sides AB, BC are equal to the two sides DE, EF respectively, and they contain equal angles ; therefore the base -^C is equal to the base DF, the'triangle ABC equal to the triangle DEF, and the remaining angle BAC equal to the remaining angle 10 EDF. [1.4I Therefore etc. rt Q. E. D. f 1 — 3. the side adjoining the equal angles, vXevpdr rV vp^ Ta7f fo-cuf 'pmUm. 99. is by hjrpothesis equal. Imhtntrnx C^, aocording to the elegant Greek idiom. iwUutiM is used for the passive of (nrcrrlBiiiu^ as iretMoi is used for the passive of rlBiifu, and io with the other oompoonds. Cf. vpoffKOffieu, ** to be added.'* The altemative method of proving this proposition, viz. by applying one triangle to the other, was very early discovered, at least so far as r^utls the case where the equal sides are adjacent to the equal angles in each. ^-NairizI gives it for this case, observing that the proof is one which he had found, but of which he did not know the author. ii 304 BOOK I [I. ,6 Proclus has the following interesting note (p. 359, 13 — 18): "Eudemiu in his geometrical history refers this theorem to Thales. For he says that» in the method by which they say that Thales proved the distance of ships in the sea, it was necessary to make use of this theorem." As, wiibrtunatdy, this information is not sufficient of itself to enable us to determine how Thales solved this problem, there is considerable room for conjecture as to his method. The suggestions of Bretschneider and Cantor agree in the assumption that the necessary observations were probably made from the top oi some tower or structure of known height, and that a right-angled triangle was used in which the tower was the perpendicular, and the line connecting the bottom of the tower and the ship was the base, as in the annexed figure, where AB is the tower and C the ship. Bretschneider {Die Geometrie und die Geometer par EukUides^ % 30) says that it was only necessary for the observer to observe the angle CAB^ and then the triangle would be completely determined by means of this angle and the known length AB. As Bretschneider says that the result would be obtained '' in a moment " by this method, it is not clear in what sense he supposes Thales to have "observed" the angle BAC. Cantor is more definite (GescK d. Math, i„ p. 145), for he says that the problem was nearly related to that of finding the Seqt from given sides. By the Seqt in the Papyrus Rhind is meant, according to the conjecture of Cantor and Eisenlohr, a number representing the ratio to one another of the lengths of certain lines in pyramids or obelisks ; sometimes it is practically equivalent to the cosine of the angle made by the sloping edge of a pyramid and the semi-diagonal of the base, sometimes to the tangent of the angle made by the perpendicular from the vertex of the pyramid on one side of the base and the line connecting the foot of that perpendicular and the centre of the base. The calculation of the Seqt thus implying a sort of theory of similarity, or even of trigonometry, the suggestion of Cantor is apparenUy that the Seqt in this case would be found from a small right-angled triangle ADE having a common angle A with ABC as shown in the figure, and that the ascertained value of the Seqt with the length AB would determine BC. This amounts to the use of the property of similar triangles; and Bretschneider's suggestion must apparenUy come to the same thing, since, even if Thales measured the angle in our sense (e.g. by its ratio to a right angle), he would, in the absence of something corresponding to a table of trigonometrical ratios, have gained nothing and would have had to work out the proportions all the same. Max C. P. Schmidt also {Kulturhistorische Beitrdge %ur Kenntnis des griechischen und rihnischen Altertums^ 1906, p. 32) similarly supposes Thales to have had a right angle made of wood or bronze with the legs graduated, to have placed it in the position ADE (A being the position of his eye), and then to have read ofi* the lengths AD, DE respectively, and worked out the length of BC by rule of three How then does the supposed use of similar triangles and their property square with Eudemus' remark about i. 26? As it stands, it asserts the equality of ttvo triangles which have two angles and one side respectively equal, and the theorem can only be brought into relation with the above explanations by taking it as asserting that, if two angles and one side of one triangle are given, the triangle is completely determined. But, if Thales I. 26] PROPOSITION 26 305 / practically Msed froportionsy as supposed, i. 26 is surely not at all the theorem which this proc^ure would naturally suggest as underlying it and being ( *' necessarily used"; the use of proportions or of similar but not equal ^ triangles would surely have taken attention altogether away from i. 26 and fixed it on vi. 4. ) For this reason I think Tannery is on the right road when he tries to find a solution using i. 26 as it stands, and withal as primitive as any recorded solution of such a problem. His suggestion (La Giomitrie gncque^ pp. 90—1) * is based on the fluminis varaiio of the Roman agrimensor Marcus Junius Nipsus and is as follows. iTo find the distance from a point ^ to an inaccessible point B, From A measure along a straight line at right angles to AB a length i^C and bisect it at Z>. From C draw CE at right angles to CA on the side of it remote from B^ and let E be the point on it which is in a straight line with B and D, Then, by 1. 26, CE is obviously equal to AB, As r^ards the equality of angles, it is to be observed that those at D are equal beotuse they are vertically opposite, and, curiously enough, Thales is expressly ^ credited with the discovery of the equality of such angles. The only objection which I can see to Tannery's ' solution is that it would require, in the case of the ship, a certain extent of fi-ee and level ground for the construction and measurements. I suggest therefore that the following may have been Thales' method. Assuming that he was on the top of a tower, he had only to use a rough instrument made of a straight stick and a cross-piece fastened to it so as to be capable of turning about the fastening (say a nail) so that it could form any angle with the stick and would remain where it was put. Then the natural thing would be to fix the stick upright (by means of a plumb-line) and direct the cross-piece towards the ship. Next, leaving the cross-piece at the angle so found, the stick could be turned round, still remaining vertical, until the cross-piece pointed to some visible object on the shore, when the object could be mentally noted and the distance ^ from the bottom of the tower to it could be subsequently measured. This ! would, by i. 26, give the distance from the bottom of the tower to the ship. i This solution has the advantage of corresponding better to the simpler and more probable version of Thales' method of measuring the height of the pyramids; Diogenes Laertius says namely (i. 27, p. 6, ed. Cobet) on the authority of Hieronymus of Rhodes (b.c 300 — 260), that he waited for this purpose until the moment when our shadows are of the same length as ourselves. Recapitulation of congruence theorems. Proclus, like other commentators, gives at this point (p. 347, 20 sqq.) a summary of the cases in which the equality of two triangles m all respects can be established. We may, he says, seek the conditions of such equality by (successively considering as hypotheses the eqiiality (i) of sides alone, (2) of angles alone, (3) of sides and angles combined. Taking (i) first, we can only establish the equality of the triangles in all respects if all three sides are respectively equal; we cannot establish the equality of the triangles by any hypothesis of class (2), not even the hypothesis that all the three angles are respectively eqiial ; among the hypotheses of class (3), the equality of one H. E. 20 3o6 BOOK I [i.a6 side and one angle in each triangle is not enough, the equality (a) ci one side and all three angles is more than enough, as is also the equali^ {i) of two sides and two or three angles, and {c) of three sides and one or two an^^ The only hypotheses therefore to be eramined firom this point of view aie the equality of (a) three sides [EucL l 8]. 03) two sides and one angle ^i. 4 proves one case of this, where the an^ is that contained by the sides which are by hypothesis equal]. (y) one side and two angles [i. 36 covers all cases]. It is curious that Proclus makes no allusion to what we call the ambiguams case^ that case namely of 09) in which it is an angle opposite to one ^tbe two specified sides in one triangle which is ecjual to the angle opposite to the equal side in the other triangle. Camerer indeed attributes to Proclus the observation that in this case the equality of the triangles caimot, unless scmie other condition is added, be asserted generally; but it would appear that Camerer was probably misled by a figure (Proclus, p. 351) which looks like a figure of the ambiguous case but is really only used to show that in l 26 the equal sides must be camspandrng sides, i.e., they must be either adjacent to the equal angles in each triangle, or opposite to corresponding equal angles, and that, e.g., one of the equd sides must not be adjacent to the two angles in one triangle, while the side equal to it in the other triangle is opposite to one of the two corresponding angles in that triangle. The ambiguous case. If two friangies have two sides equal io two sides respecHveiy^ and if ike angles opposite to one pair of equal sides he also equals then will ike angles opposite the other pair of equal sides be either equal or suppUmmtary ; emd^ in the former case^ the triangles will be equal in all respects. Let ABC, DEFht two triangles such that AB is equal to DE, and AC to DE, while the angle ABC is equal to the angle DEF\ it is required to prove that the angles ACB, DFE are either equal or supplementary. ^ A D Now (i), if the angle BAC be equal to the angle EDF, it follows, since the two sides AB^ AC Site equal to the two sides !?£, 27^ respectively, that . the triangles ABC, DEFdj^ equal in all respects, [i. 4] and the angles ACB, DFE are equal. (2) If the angles BAC, EDF be not equal, make the angle EDG (on the same side of ED as the angle EDF) equal to the angle BAC Let EF, produced if necessary, meet DG in G. Then, in the triangles ABC, DEG, the two angles BAC, ABC are equal to the two angles EDG, DEG respectively, and die side AB is equal to the side DE ; J I. 26, 27] PROPOSITIONS 26, 27 307 therefore the triangles ABC^ DEG are equal in all respects, [i. 26] so that the side ^C is equal to the side DG^ and the angle ACB is equal to the angle DGE. Again, since AC is eqiial to DF9& well as to DG^ DF\& equal to DG, and therefore the angles DFG^ DGF^xt, equal. But the angle DFE is supplementary to the angle DFG\ and the angle DGF^N^s proved equal to the angle ACB; therefore the angle I?F£ is supplementary to the angle ACB. If it is desired to avoid the ambiguity and secure that the triangles may be congruent, we can introduce the necessary conditions into the enunciation, on the analogy of Eucl. vi. 7. Jf (tvo triangles have two sides of the one equal to two sides of the other ) respectively^ and the angles opposite to a pair of equal sides equals then, if the i angles opposite to the other pair of equal sides are both acute^ or hoth obtuse^ or if one of them is a right angle^ the two triangles are equal in all respects. (The proof of the three cases (by reductio cui absurdum) was given by Todhunter. Proposition 27. If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines mill be parallel to one another. % For let the straight line EF falling on the two straight 5 lines AB, CD make the alternate angles AEF, EFD equal ^ to one another ; I say that AB is parallel to CD. For, if not, AB, CD when pro- duced will meet either in the direction \o of B, D or towardis A, C. Let them be produced and meet, in the direction of B, Z>, at G. * Then, in the triangle GEF, the exterior angle A EF is equal to the interior and opposite 15 angle EFG : which is impossible. [i. 16] Therefore AB, CD when produced will not meet in the direction of B, D. Similarly it can be proved that neither will they meet o towards A, C 20 — 2 i 3o8 BOOK I [i. «7 But straight lines which do not meet in either direction are parallel ; [Dcf. 23] therefore AB is parallel to CD. Therefore etc. Q. E. D. I. falling on two straight lines, 4,% Mo MtU% kiarifmntti^ the phnse being the flune as that used in Post. 5, meaning a trtmsversml* 3. the alternate angles, oi ^miXXAI Twrku. Produt fp. ^57, 9) expfauns that Eadid uses the word aitemate (or, more exactly, aiiimatefy^ ^roXXd^ m two connexions, (i) of a certain transformation of a proportion, as in Book V. and the arithmetical Books, (s) as here, of certain of the angles formed by parallels with a straight line crossing them. AlUnmU angles are, according to Euclid as interpreted by Prodns, those which are not on the same side of the transveruil, and are not adjacent, but are separated by the transversal, both being within the parallels but one ** above ** and the other ** below." The meaning is natoru enough if we imagine the four internal angles to be taken in cyclic order and aH^nmU angkf to be any two of them not successive but separated by one angle of the four.' 9. in the direction of B, D or towards A, C, literally ** towards the farts BtD ot towards A, C," M rii B, A fidpti 41 M rii A, T. With this proposition b^;ins the second section of the first BodL Up to this point Eudid has dealt mainly with triangles, their constmcdcm and their properties in the sense of the relation of their parts, the tides and angles, to one another, and the comparison of different triangles in respect oi their parts, and of their area in the particular cases where they are congruent The second section leads up to the third, in which we pass to rdations between the areas of triangles, parallelograms and squares, the special feature being a new conception of equality of areas, equality not dependent on congruence. This whole subject requires the use of parallels. Consequently the second section beginning at i. 27 establishes the theory of parallels, introduces the cognate matter of the equality of the sum of Uie angles of a triangle to two right angles (i. 32), and ends with two propositions forming the ^ transition to the third section, namely i. 33, 34, which introduce the parallelo- y gram for the first time. Aristotle on parallels. ] We have already seen reason to believe that Euclid's personal contribution ' to the subject was nothing less than the formulation of the famous Postulate 5 (see the notes on that Postulate and on Def. 23), since Aristotle indicates that the then current theory of parallels contained a petiiio prindpii^ and . ! presumably therefore it was EucUd who saw the defect and proposed the ^ remedy. But it is clear that the propositions i. 27, 28 were contained in earlier text-books. They were familiar to Aristotle, as we may judge from two j interesting passages. I (i) In Anal. Post i. 5 he is explaining that a scientific demonstration must not only prove a fact of every individual of a class (#car& wavro^) but ' must prove it primarily and generally true (irp«;^ov #ca^<^Xov) of the whole of the class as one ; it will not do to prove it first of one part, then of another . part, and so on, until the class is exhausted. He illustrates this (74 a 13 — 16) I by a reference to parallels : '' If then one were to show that right (angles) do not meet, the proof of this might be thought to depend on the fact diat this is true of all (pairs of actual) right angles. But this is not so, inasmuch as the result does not follow because (the two angles are) eqtial (to two right r 1. 27, 28] PROPOSITIONS 27. 28 309 angles^ in the particular way [i.e. because each is a right angle], but by virtue i of their being equal (to two right angles) in any way whatever [i.e. because I the sum only needs to be equal to two right angles, and the angles themselves may vary as much as we please subject to this]." ' (2) The second passage has already been quoted in the note on Def. 23 : ' *' there is nothing surprising in different hypotheses leading to the same false ^ conclusion ; e.g. the conclusion that parallels meet might equally be drawn I from either of the assumptions (a) that the interior (angle) is greater than the J exterior or {b) that the sum of the angles of a triangle is greater than two I right angles" (Anal. Prior, 11. 17, 66 a 11 — 15). \ I do not quite concur in the interpretation which Heiberg places upon these passages (Mathematisches tu AristoteleSy pp. 18 — 19). He says, first, that the allusion to the "interior angle" being "greater than the exterior" in I the second passage shows that the reference in the first passage must be to Eucl. I. 28 and not to i. 27, and he therefore takes the words ^i wSl Strot in the first passage (which I have translated " because the two angles are equal to two right angles in the particular way ") as meaning " because the angles, viz. the exterior and the interior^ are equal in the particular way." He also takes oi ^p^ai ov (rvfixurrovcri (which I have translated " right angles do not meet," an expression quite in Aristotle's manner) to mean "perpendicular straight lines do not meet " ; this is very awkward, especially as he is obliged to supply angles with uroi in the next sentence. But I think that the first passage certainly refers to i. 28, although I do not think that the alternative {a) in the second passage suggests it. This alternative may, I think, equally with the alternative {b) refer to i. 27. That proposition is proved by rcductio ad absurdum based on the fact that, if the straight lines do meet, they must form a triangle^ in which case the exterior angle must be greater than the interior (while according to the hypothesis these angles are equal). It is true that Aristotle speaks of the hypothesis that the interior anele is greater than the exterior ; but after all Aristotle had only to state some incorrect hypothesis. It is of course only in connexion with straight lines meetings as the hypothesis in 1. 27 makes them, that the alternative {b) about the sum of the angles of a triangle could come in, and alternative {a) implies alternative (b). It seems clear then from AristoUe that i. 27, 28 at least are pre-Euclidean, ^d that it was only in i. 29 that Euclid made a change by using his Postulate. De Morgan observes that i. 27 is a logical equivalent of i. 16. Thus, if A means "straight lines forming a triangle with a transversal," B "straight lines making angles with a transversal on the same side which are together less than two right angles," we have All ^ is ^, and it follows logically that All not-^ is not-^. Proposition 28. If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, Jhe straight lines will be parallel to one another. 3IO BOOK I [i. 38 For let the straight line EF falling on the two straight lines AB, CD make the exterior angle EGB equal to the interior and opposite angle GHD, or the interior angles on the same side, namely BGH, GHD, equal to two right angles; : I say that AB is parallel to CD. For, since the angle EGB is equal to the angle GHD, while the angle EGB is equal to the angle AGH, [i. 15] the angle AGH is also equal to the angle GHD\ and they are alternate ; therefore AB is papdlel to CD. [i. 27] Again, since the angles BG^.JCrlip^zxt^ equal to two right angles, and the angles AGH, BuH are also equal to two right angles, [i. 13] the angles AGH, BGH are equal to the angles BGH, GHD. Let the angle BGH be subtracted from each ; therefore the remaining angle AGH is equal to the remaining angle GHD\ and they are alternate ; therefore AB is parallel to CD. [i. 27] Therefore etc. j Q. E. D. One criterion of parallelism, the equality of alternate angles, is given in I. 37 ; here we have two more, each of which is easily reducible, and is actually reduced, to the other. Produs observes (pp. 358 — 9) that Euclid could have stated six criteria as well as three, by using, m addition, other pairs of angles in the figure (not adjacent) of which it could be predi- cated that the two angles are equal or that their sum is ec^ual to two right angles. A natural division is to consider, first the pairs which are on the same side of the transversal, and secondly the pairs which are on different sides of it Taking (i) the possible pairs on the satne side, we may have a pair consisting of (a) two internal angles, viz. the pairs (BGH, GHD) and (AGH, GHC) ; (i) two external angles, viz. the pairs (EGB, DHF) and {EGA, CHF)\ (i) one external and one internal angle, viz. the pairs lEGB^ GHD\ (FHD, HGS), {EGA, GHC) and {FHC, HGA). I. 28, 29] PROPOSITIONS 28, 29 311 And (2) the possible pairs on different sides of the transversal may consist respectively of (fl) two internal angles, viz. the pairs {AGH, GHD) and (CZTG, HGE)\ \b) two external angles, viz. the pairs {AGE, DHF) and {EGB, CHF)\ (f) one external and one internal, viz. the pairs {AGE, GHD\ lEGB, GHC\ {FHC, HGB) and {FHD, HGA). The angles are equal in the pairs (i) (r), (2) (a) and (2) {b\ and the sum is equal to two right angles in the case of the pairs (i) {a\ (i) {p) and (2) (r). For his criteria Euclid selects the cases (2) (a) [i. 27J and (i) {c\ (i) (a) [i. 28], leaving out the other three, which are of course equivalent but are not quite so easily expressed From Proclus' note on i. 28 (p. 361) we learn that one Aigeias (? Aineias) of Hierapolis wrote an epitome or abridgment of the Elements, This seems to be the only mention of this editor and his work; and they are only mentioned as having combined Eucl. i. 27, 28 into one proposition. To do this, or to make the three hypotheses the subject of three separate theorems, would, Proclus thinks, have been more natural than to deal with them, as Euclid does, in two propositions. Proclus has no suggestion for explaining Euclid's arrangement unless the ground were that i. 27 deals with angles on different sides, i. 28 with angles on one and the same side, of the transversal. But may not the reason have been one of convenience, namely that the criterion of i. 27 is that actually used to prove parallelism, and is moreover the basis of the construction of parallels in i. 31, while i. 28 only reduces the other two hypotheses to that of i. 27, so that precision of reference, as well as clearness of exposition, is better secured by the arrangement adopted ? Proposition 29. A straight line falling on parallel straight lines makes \ the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. 5 For let the straight line EF fall on the parallel straight - lines AB, CD ; I say that it makes the alternate angles AGH, GHD equal, the ekterior angle EGB equal to the interior and opposite angle GHD, and the interior angles on the same 10 side, namely BGH, GHD, equal to two right angles. For, if the angle AGH is unequal to the angle GHD, one of them is greater. Let the angle AGH be greater. 15 Let the angle BGH be added to each ; therdfore the angles AGH, BGH are greater than the angles BGH, GHD. 1' 31 a BOOK I [h 29 «> But the angles AGIf, BGH are equal to two right angles; therefore the angles BGH, GHD are less than two right angles. But straight lines produced indefinitely from angles less than two right angles meet ; [Post. 5] 25 therefore AB, CD, if produced indefinitely, will meet ; but they do not meet, because they are by hypothesis parallel. Therefore the angle AGH is not unequal to the angle GHD, and is therefore equal to it JO Again, the angle AGH is equal to the angle EGB ; [i. 15] therefore the angle EGB is also equal to the angle GHD. [C. N. i] Let the angle BGH be added to each ; therefore the angles EGB, BGH are equal to the js angles BGH. GHD. [C. N. a] But the angles EGB, BGH are equal to two right angles; P-I3] therefore the angles BGH, GHD are also equal to two right angles. Therefore etc. q. e. d. 13. straight lines produced indefinitely from angles less than two right angles, al M dr* IkaffoiufWf if h^ 6p$C» iKfioKKhfUPot, e/f dw€ipow cvftwlwrouavt a variation from the more explicit language of Postulate 5. A £ood deal is left to be understood, namely that the straight lines begin from pointo at which they meet a thmsversal, and make with it internal angles on the same side the sum of which is equal to two right angles. a6. because they are by hjrpothesis parallel, literally ** because they are supposed parallel," Ml t6 ra^M^i^Xovf uMls ihrmrfSytfoi. Proof by " Plasrfair's " axiom. If, instead of Postulate 5, it is prefen-ed to use '' Playfaiifs " axiom in the proof of this proposition, we proceed thus. To prove that the alternate angles AGff, v ^l GIfD are equal. ^^ ,^- "" p If they are not equal, draw another straight K- \ line I^L through G making the angle XGIf q Jv^ o equal to the angle GIfD. \ Then, since the angles KGIf, GIfD are equal, ^ JTZ is parallel to CD. [i. 27] Therefore ttvo sira^ht lines KL» AB intersecting at G are both parallel to the straight line CD : which is impossible (by the axiom). Therefore the angle ^C?^ cannot but be equal to the angle GHD. The rest of the proposition follows as in Euclid. I 1.291 PROPOSITION 29 313 Proof of Euclid's Postulate 5 from " Plasrfair's " axiom. Let ABy CD make with the transversal EF the angles AEF^ EFC together less than two right angles. To prove that AB^ CD meet towards A^ C p _— — -B Through E draw GH making with EF the angle Q.j^^;:::::;.^^-^^^^^^^— --H GJS/' equal (and alternate) to the angle EFD, ^ \ Thus G-^is parallel to CD, [i. 27] \ Then (i) AB must meet CD in one direction or ^ F the other. For, if it does not, AB must be parallel to CD\ hence we have two straight lines AB^ GH intersecting at E and both parallel to CD : which is impossible. ^j2^ v^la-^-^. juy^^ .^v^ Therefore AB^ CD must meet. (2) Since ABy CD meet, they must form a triangle with EF, But in any triangle any two angles are together less than two right angles. Therefore the angles AEF^ EFC (which are less than two right angles), and not the angles BEF^ EFD (which are together greater than two right angles, by i. 13), are the angles of the triangle ; that is, EAy FC meet in the direction of A^ C, or on the side of EF on which are the angles together less than two right angles. The usual course in modem text-books which use " Playfair's " axiom in lieu of Euclid's Postulate is apparently to prove i. 29 by means of the axiom, and then Euclid's Postulate by means of i. 29. De Morgan would introduce the proof of Postulate 5 by means of "Playfair's" axiom before i. 29, and would therefore apparently prove i. 29 as Euclid does, without any change. As between Euclid's Postulate 5 and " Playfair's " axiom, it would appear that the tendency in modem text-books is rather in favour of the latter. Thus, to take a few noteworthy foreign writers, we find that Rausenberger stands almost alone in using Euclid's Postulate, while Hilbert, Henrici and Treutlein, Rouch^ and De Comberousse, Enriques and Amaldi all use " Playfair's " axiom. Yet the case for preferring Euclid's Postulate is argued with some force by Dodgson (Euclid and his modem Rivals^ pp. 44—6). He maintains (i) that :' Playfair's" axiom in fact involves Euclid's Postulate, but at the same time I involves tnare than the latter, so that, to that extent, it is a needless strain on the faith of the learner. This is shown as follows. Given AB, CD making with EF\}^t, angles AEF, EFC together less than two right angles, draw GH through E so that the angles GEF, EFC are together equal to two right angles. Then, by i. 28, GH, CD are ''separational." We see then that any lines which have the property (a) that they make with a transversal angles less than two right angles have also the property {fi) that one of them intersects a straight line which is '' separational " from the other. Now Playfair's axiom asserts that the lines which have property (fi) meet if produced : for, if they did not, we should have two intersecting straight lines both '' separational " from a third, which is impossible. We then argue that lines having property (a) meet because lines having property (a) are lines having property (fi). But we do not know, until we have proved i. 29, that all pairs of lines having property (fi) have also property 314 BOOK I [i. 29, 30 (a). For anything we know to the contiary, class (fi) mmy be greater than class (a). Hence, if you assert anything of dass (/S), the logical ^ect is more extensive than if you assert it of dass (a) \ for you assert it, not only of that portion of class (fi) which is known to be included in dass (a), but abo of the unknown (but possibly existing) portion which is ff^/ so induded. < (2) Eudid's Postulate pjuU before the beginner clear and posUm con- ceptions, a pair of straight lines, a transversal, and two angl^ together less than two right angles, whereas " Playfair's " axiom requires him to reaEse a pair of straight lines which never meet though produced to infinity : a mgatioe conception which does not convey to the mmd any clear notion of therdative position of the lines. And (p. 68) Eudid's Postulate gives a direct criterion for judging that two straight hnes meet, a criterion which is constantly reqdied, e^. in I. 44. It is true that the Postulate can be deduced from ^Playftir's'' axiom, but editors frequently omit to deduce it, and then taddy assume it afterwards : which is the least justifiable course of all Proposition 30. Straight lines parallel to the same straight line are also parallel to one another. Let each of the straight lines AB^ CD be parallel to EF\ I say that AB is also parallel to CD. 5 For let the straight line GK fall upon them. Then, since the straight line GK has fallen on the parallel straight lines AB, EF, o the angle AGK is equal to the angle GHF. [l 29] j Again, since the straight line GK has fallen on the parallel I straight lines EF, CD, - the angle GHF is equal to the angle GKD. [i. 29] 5 But the angle AGK "w^s also proved equal to the angle GHF\ therefore the angle AGK is also equal to the angle GKD ; [C. N. il and they are alternate. o Therefore AB is parallel to CD. Q. E. D. «o. The usual conclusUn in general terms (" Therefore etc'*) repeatiiig the ennnciatioii is, curiously enough, wanting at the end of this proposition. The proposition is, as De Morgan points out, the logical equivalent of ''Playfoir's" axiom. Thus, if X denote "pairs of straight lines intersecting one 30. 3'] PROPOSITIONS 29—31 3>S another," K^' pairs of straight lines parallel to one and the same straight line," we have No X is K, and it follows logically that No Y is X. De Morgan adds that a proposition is much wanted about parallels ^or perpendiculars) to two straight lines respectively making the same angles with one another as the latter do. The proposition may be enunciated thus : If the sides of one angle be respectively (i) parallel or (2) perpendicular to the sides of another ang^^ the two angles are either equal or supplementary. (i) Let DE be parallel to AB and GEF parallel to BC. To prove that the angles ABC^ DEG are equal and the angles ABC^ JDEF supplementary. Produce DE to meet BC in H, Then [i. 29] the angle DEG is equal to the angle DHC, and the angle ABC is equal to the angle DHC Therefore the angle DEG is equal to the angle ABC\ whence also the angle DEF is supplementary to the angle ABC (2) Let ED be perpendicular to AB, and GEF perpendicular to BC, To prove that the angles ABC, DEG are equal, and the angles ABC, DEF supplementary. Draw Ejy at right angles to ED on the side of it opposite to B, and (Uaw EG at right angles to EF on the side of it opposite to B, Then, since the angles BDE, DED, being right angles, are equal, ED is parallel to BA. [i. 27] Similarly EG' is parallel to BC Therefore [Part (i)] the angle DEG is equal to the angle ABC But, the right angle DED being equal to the right angle GEG, if the common angle GEuht subtracted, the angle DEG is equal to the angle DEG , Therefore the angle DEG is equal to the angle ABC\ and hence the angle DEF is supplementary to the angle ABC Proposition 31. Through a given point to draw a straight line parallel to a given straight line. Let A be the given point, and BC the given straight line; thos it is required to draw through the point A a straight line parallel to the straight line Bi 3i6 BOOK I [i. 31, 3a Let a point D be taken at random on BC, and let AD be joined; on the straight line DA, and at the point A on it, let the ^ — angle DAE be constructed equal / to the angle ADC [i. 23] ; and let the straight straight ine AF be produced in a ine with EA. I Then, since the straight line AD falling on the two straight lines BC, EF has made the alternate angles EAD, ADC equal to one another, therefore EAF is parallel to BC. [i. «7] Therefore through the given point A the straight line EAF has been drawn parallel to the given straight line BC. Q. E, F, Proclus rightly remarks (p. 376, 14 — 20) that, as it is implied in i. 12 that only one perpendicular can be drawn to a straight line from an external point, so here it is implied that only one straight line can be drawn through a point parallel to a given straight line. The construction, be it observed, depends only upon i. 27, and might therefore have come directly after that proposition, ^^y then did Eucbd postpone it until after i. 29 and L 30? Presumably because he considered it nec^sary, before giving the constructioii, to place beyond all doubt the fact that only one such parallel am be drawn. Proclus infers this fact from i. 30 ; for, he says, if two straight lines could be drawn through one and the same point parallel to the same straight line, the two straight lines would \^ parallel, though intersecting at the given point : which is impossible. I think it is a fair inference that Euclid would have considered it necessary to justify the assumption that only one parallel can be drawn by some such argument, and that he deliberately determined that his own assumption was more appropriate to be made the subject of a Postulate than tfie assumption of the uniqueness of the parallel Proposition 32. In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles^ and the three interior angles of the triangle are equal to two right angles. Let ABC be a triangle, and let one side of it BC be produced to D ; I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles of the triangle ABC^ BCA, CAB are equal to two right angles. I, 32] PROPOSITIONS 31. 32 317 For let CE be drawn through the point C parallel to the straight line AB. [i. 31] Then, since AB is parallel to CE, and AC has fallen upon them, the alternate angles BAC, ACE are equal to one another. [i. 29] Again, since AB is parallel to CE, and the straight line BD has fallen upon them, the exterior angle ECD is equal to the interior and opposite angle ABC. [i. 29] But the angle ACE was also proved equal to the angle BAC\ therefore the whole angle ACD is equal to the two interior and opposite angles BAC, ABC Let the angle ACB be added to each ; therefore the angles ACD, ACB are equal to the three angles ABC, BCA, CAB. But the angles ACD, ACB are equal to two right angles; P- 13] therefore the angles ABC, BCA, CAB are also equal to two right angles. Therefore etc, Q. E. D. This theorem was discovered in the very early stages of Greek geometry. What we know of the history of it is gathered from three allusions found in Eutocius, Proclus and Diogenes Laertius respectively. I. Eutocius at the beginning of his commentary on the Conies of ''Apollonius (ed. Heiberg, Vol. 11. p. 170) quotes Geminus as saying that *'the ancients (oc ^x^^ot) investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle, and later geometers demonstrated the general theorem to the effect that in any triangle the three interior angles are equal to two right angles." 2. Now, according to Proclus (p. 379, 2 — 5), Eudemus the Peripatetic refers the discovery of this theorem to the Pythagoreans and gives what he affirms to be their demonstration of it This demonstration will be given below, but it should be remarked that it is general, and therefore that the "later geometers" spoken of by Geminus were presumably the Pythagoreans, whence it appears that the ''ancients" contrasted with them must have belonged to the time of Thales, if they were not his Egyptian instructors. 3. That the truth of the theorem was known to Thales might also be inferred from the statement of Pamphile (quoted by Diogenes Laertius, L 24 — 5, p. 6, ed. Cobet) that "he, having learnt geometry from the 3i8 BOOK I [L39 Egyptians, was the first to inscribe a right-angled triang^ in a aide and sacrificed an ox" (on the strength of it) ', in odier words, he discovered that the angle in a semicircle is a right angle. No doubt, when this fact wu once discovered (empinca/fy^ say), the consideration of the two isosceles triangles having the centre for vertex and the sides of the right angle for bases respectively, with the help of the theorem of EucL i. 5, also known to Thales, would easily lead to the conclusion that the sum of the angles of a right-angled triangle is equal to two right angles, and it could be readfly inferred that the angles of any triangle were likewise equal to two right angles (by resolving it into two right-angled triangles). But it is not easy to see how the property of the angle in a semicircle could h^prwed except (in the revise order) by means of the equality of the sum of the angles of a right-an^ii triangle to two right angles ; and hence it is most natural to suppose, with Cantor, that Thales proved it (if he did prove it) practically as Euclid does in III. 31, i.e. by means of i. 32 as applied to righi-angled triangles at all events. If the theorem of i. 32 was proved before Thales' time, or by Thales himself, by the stages indicated in the note of Geminus, we may be satisfied that the reconstruction of the argument of the older proof by Hankd (pp. 96 — 7) and Cantor (i„ pp. 143 — 4) is not far wronj;. First, it must have be^sn observed that six angles equal to an angle of an eqmlateral triangle would, if placed adjacent to one another round a common vertex, fill up the whde space round that vertex. It is true that Produs attributes to the Pythafpreans the general theorem that only three kinds of regular polygons, the equilateral trianf^le, the square and the regular hexagon, can fill up the entire space round a point, but the practical Imowled^ge that equilateral triangles have this property could hardly have escaped the Egyptians, whether they made floors with tiles in the form of equilateral triangles or regular hexagons (Allman, Greek Geometry from Thales to Euciid, p. 13) or joined the ends of adjacent radii of a figure like the six-spoked wheel, which was their common form of whed fi'om the time of Ramses II. of the nineteenth Dynasty, say 1300 b.c (Cantor, i,, p. 109). It would then be dear that six angles equal to an angle of an equilateral triangle are equal to four right angles, and therefore that the three angles of an equilateral triangle are equal to two right angles. (It would be as clear or clearer, from observation of a square divided into two triangles by a diagonal, that an isosceles right-angled triangle has each of its equal angles equal to half a right angle, so that an isosceles right-angled triangle must have the sum of its angles equal to two right angles.) Next, with leffxA to the equilateral triangle, it could not fail to be observed that, if AD were drawn from the vertex A perpendicular to the base BC, each of the two right-angled triangles so formed would have the sum of its angles equal to two right y angles ; and this would be confirmed by completing the / rectanp;le ADCE, when it would be, seen that the rectangle / (with Its angles equal to four right angles) was divided by / its diagonal into two equal triangles, each of which had 6 the sum of its angles equal to two right angles. Next it would be inferred, as the result of drawing the diagonal of any rectangle and observing the equality of the triangles forming the two halves, that the sum of the angles of any right-angled triangle is equal to two right angles, and henoe (the two congruent right-angled triangles being then pla^ so as to form one isoscdes triangle) that the same is true of any isosceles triangle. Only the last step remained, namely that of observing that any triangle could be regurded as the half of a rectangle (drawn as indicated in the next figure), or L32] PROPOSITION 32 319 simply that any triangle could be divided into two right-angled triangles, I whence it would be inferred that in general the I sum of the angles of any triangle is equal to two f --^^^p;— • ; I right angles. j X \ ^s^^^ j Such would be the probabilities if we could \ X \. i absolutely rely upon the statements attributed to \,^__ ! ^J Pamphile and Geminus respectively. But in fact there is considerable ground for doubt in both cases. 1. Pamphile's story of the sacrifice of an ox by Thales for joy at his discovery that the angle in a semicircle is a right angle is too suspiciously like the similar story told with reference to Pythagoras and his discovery of the theorem of Eucl. i. 47 (Proclus, p. 426, 6 — 9). And, as if this were not enough, Diogenes Laertius immediately adds that '* others, among whom is Apollodorus the calculator (d XoytorcKo^), say it was Pythagoras" (sc. who *' mscribed the right-angled triangle in a circle "). Now Pamphile lived in the reign of Nero (a.d. 54 — 68) and therefore some 700 years after the birth of Thales (about 640 b.c). I do not know on what Max Schmidt bases his statement {Kulturhistorische Beitrdge%ur Kenntnis des griechischen und romischm AUertums^ 1906, p. 31) that "other, much older^ sources name Pythagoras as the discoverer of the said proposition," because nothing more seems to be known of Apollodorus than what is stated here by Diogenes Laertius. But it would at least appear that Apollodorus was only one of several authorities who attributed the proposition to Pythagoras, while Pamphile is alone mentioned as referring it to Thales. Again, the connexion of Pythagoras with the investigation of the right-angled triangle makes it a priori more likely that it would be he who would discover its relation to a semicircle On the whole, therefore, the attribution to Thales would seem to be more than doubtful. 2. As regards Geminus' account of the three stages through which the proof of the theorem of 1. 32 passed, we note, first, that it is certainly not confirmed by Eudemus, who referred to the Pythagoreans the discovery of the theorem that the sum of the angles of any triangle is equal to two right angles and says nothing about any gradual stages by which it was proved. Secondly, it must be admitted, I think, that in the evolution of the proof as reconstructed by Hankel the middle stage is rather artificial and unnecessary, ^since, once it is proved that any right-angled triangle has the sum of its angles ecjual to two ri^ht angles, it is just as easy to pass at once to any scalene triangle (which is decomposable into two unequal right-angled triangles) as to the isosceles triangle made up of two congruent right-angled triangles. Thirdly, as Heiberg has recently pointed out (Mathematisches zu Aristoteles^ p. 20), it is quite possible that the statement of Geminus from beginning to end is simply due to a misapprehension of a passage of Aristotle {Anal, Post. i. 5, 74 a 25). Aristotle is illustrating his contention that a property is not scientifically proved to belong to a class of things unless it is proved to belong primarily {wfHarov) and generally (iraMXou) to the whole of the class. His first illustration relates to parallels making with a transversal angles on the same side together equal to two right angles, and has been quoted above in the note on I. 27 (pp. 308—9). His second illustration refers to the transformation of a proportion altemando^ which (he says) ''used at one time to be proved separately " for numbers, lines, solids, and times, although it admits of being proved of all at once by one demonstration. The third illustration is : " For the same reason, even if one should prove (ovS* av rvi Sciif^) with reference to 320 BOOK I [1.32 each (sort of) triangle, the equilateral, scalene and isosceles, separateljr, that each has its angles equal to two right angles, either by one proof or by different proofs, he does not yet know that t?u trian^^ i.e. the triangle in gtmrai^ has Its angles equal to two right angles, except in a sophistical sense, even though there exists no triangle other than triangles of the kinds mentioned. For he knows it, not qu& triangle, nor of evtry triansle, except in a numerical sense (car apiO/ior); he does not know it noHonally (icar cISo«) of every triangle^ even though there be actually no triangle which he does not know." The difference between the phrase *' used at one time to be proved " in the second illustration and '' if any one should prove " in the third appears to I indicate that, while the former referred to a historical fact, the latter does not; the reference to a person who should prove the theorem of i. 32 for the three kinds of triangle separately, and then claim that he had proved it generally, i| states a purely hypothetical case, a mere illustration. Yet, coming after the j historical Ibxx stated in the preceding illustration, it might not unnaturally give the impression, at first sight, that it was historicsil too. On the whole, therefore, it would seem that we cannot safely go behind the dictum of Eudemus that the discovery and proof of the theorem of i. 32 in all its generality were Pythagorean. This does not however preclude its having been discovered by stages such as those above set out after Hankel and Cantor. Nor need it be doubted that Thales and even his Egyptian instructors had advanced some way on the same road, so far at all events as to see that in an equilateral triangle, and in an isosceles right-angled triangle, the sum of the angles is equal to two right angles. The Pythagorean proof. This proof, handed down by Eudemus (Proclus, p. 379, 2 — 15), is no elegant than that given by EucUd, and is a natural development from the last figure in the recon- structed argument of Hankel. It would be seen, after the theory of parallels was added to geometry, that the actual drawing of the perpendicular and the complete rectangle on BC as base was unnecessary, and that the parallel to BC through A was all that was required. Let ABC be a triangle, and through A draw DE parallel to BC [1. 31] Then, since BC^ DE are parallel, the alternate angles DAB^ ABCzit equal, [i. 29] and so are the alternate angles EAd ACB also. Therefore the angles ABC^ ACB are together equal to the angles DAB, EAC Add to each the angle BAC; therefore the sum of the angles ABC, ACB, BAC is equal to the sum of the angles DAB, BAC, CAE, that is, to two right angles. Euclid's proof pre-Euclidean. The theorem of i. 32 is Aristotle's favourite illustration when he wishes to refer to some truth generally acknowledged, and so often does it occur that it is often indicated by two or three words in themselves hardly intelli^ble, e.g. TO fivcrlv ^p^ois (Ana/, Post, i. 24, 85 b 5) and vwapxti wavrl rpiyiiytf ro fivo {Md, 85 b 11). One passage (Metaph. 1051 a 24) makes it clear, as Heiberg (op,*dt. I. 32] PROPOSITION 32 321 p. 19) acutely observes, that in the proof as Aristotle knew it Euclid's construction was used. "Why does the triangle make up two right angles? Because the angles about one point are equal to two right angles. If then the parallel to the side had been drawn up (avrJKro), the fact would at once have oeen clear from merely looking at the figure." The words "the angles about one point" would equally fit the Pythagorean construction, but "drawn upwards " applied to the parallel to a side can only indicate Euclid's. Attempts at proof independently of parallels. The most indefatigable worker on these lines was Legendre, and a sketch of his work has been given in the note on Postulate 5 above. One other attempted proof needs to be mentioned here because it has ^ found much favour. I allude to Thibaut's method. This appeared in Thibaut's Grundriss der reinen Mathematik, Gottingen (2 ed. 1809, 3 ed. 181 8), and is to the following effect Suppose CB produced to D^ and let BD (produced to any necessary extent either way) revolve in one direction (say clockwise) first about B into the position BA^ then about A into the position oi AC produced both ways, and lastly about C mto the position CB produced both ways. The argument then is that the straight line BD has revolved through the sum of the three exterior angles of the triangle. But, since it has at the end of the revolution assumed a position in the same straight line with its original position, it must have revolved through four right angles. Therefore the sum of the three exterior angles is equal to four right angles; frx)m which it follows that the sum of the three angles of the triangle is equal to two right angles. But it is to be observed that the straight line BD revolves about different points in it^ so that there is translation combined with rotatory motion, and it is necessary to assume as an axiom that the two motions are independent, and ^therefore that the translation may be neglected. Schumacher (letter to Gauss of 3 May, 1831) tried to represent the rotatory motion graphically in a second figure as mere motion round a point ; but Gauss (letter of 17 May, 1831) pointed out in reply that he really assumed, without proving it, a proposition to the effect that " If two straight lines (i) and (2) which cut one another make angles A\ A" with a straight line (3) cutting both of them, and if a straight line (4) in the same plane is I likewise cut by (i^ at an angle A\ then (4) will be cut by (2) at the angle A\ But this proposition not only needs proof, but we may say that it is, in I essence, the very proposition to be proved" (see Engel and Stackel, Die Theorie der ParaUelHnien von Euklid bis auf Gauss^ 1895, P* ^3^)* How easy it is to be deluded in this way is plainly shown by Proclus' attempt on the same lines. He says (p. 384, 13—21) that the truth of the theorem is borne in upon us by the help of " common notions " only. " For, if we conceive a straight line with two perpendiculars drawn to it at its ex- tremities, and if we then suppose the perpendiculars to (revolve about their feet and) approach one another, so as to form a triangle, we see that, H. E. 21 322 BOOK I [i. 32, 33 to the extent to which they converge, they diminish the right an^es wtidb, they made with the straight line, so that the amomit taken from the rifjtxt angles is also the amount added to the vertical angle of the- triangle, and the three angles are necessarily made ec^ual to two right angles." But a moment^* I reflection shows that, so far from bemg founded on mere " common notiooa," the supposed proof assumes, to begin with, that, if the perpendiculars ap- proach one another ever so litde, they will then form a triangle immediately, i.e., it assumes Postulate 5 itself; and the fact about the vertical aiigle can only be seen by means of the equality of the alternate angles exhibited by drawing a perpendicular from the vertex of the triangle to the base, Le. KparaiMto eitner of the original perpendiculars. Extension to polygons. The two important corollaries added to i. 32 in Simson's edition are given by Proclus ; but Proclus' proof of the first is different from, and pertiaps somewhat simpler than, Simson's. 1. The sum of the interior anf^lts of a convex reetilifieai figure is equal to twice as many right angles as the figure has sides, less four. For let one angular point A be joined to all the other angular points with which it is not con- nected already. The figure is then divided into triangles, and mere inspection shows (i) that the number of triangles is two less than the number of sides in the figure, (2^ that the sum of the angles of all the ^ triangles is equal to the sum of sdl the interior angles of the fi^re. | Smce then the sum of the angles of each triangle is equal to two right angles, the sum of the interior angles of the figure is equal to 2 (/r-2) right angles, I Le. {2n - 4) right angles, where n is the number of sides in the figure. \ 2. The exterior angles of any convex rectilineal figure are together equal to four right cutgles. For the interior and exterior angles together are equal to 2n right angles, where n is the numb^ of sides. And the intenor angles are together equal to (2ff-4) right angles. Therefore the exterior angles are together equal to (bur ri^ht angles. This last property is already quoted by Aristotle as true of all rectilineal figures in two passages (Anal, Ast. I. 24, 85 b 38 and 11. 17, 99 a 19). Proposition 33. TAe stra^kt iines joining equal ami parallel straight lines (at the extremities which are) in the same directions {respectively) are themselves also equcU and parallel. Let AB, CD be equal and parallel and let the straight 5 lines AC^ BD join them (at the extremities which are) in the same directions (respectively) ; I. 33i 34] PROPOSITIONS 32—34 323 ; I say that AC, BD are also equal and parallel. Let BC be joined. Then, since AB is parallel to CD, 10 and BC has fallen upon them, the alternate angles ABC, BCD are equal to one another. [i. 29] And, since AB is equal to CD, and BC is common, 15 the two sides AB, BC are equal to the two sides DC, CB ; and the angle ABC is equal to the angle BCD ; therefore the base -^C is equal to the base BD, and the triangle ABC is equal to the triangle DCB, and the remaining apgles will be equal to the remaining angles 20 respectively, namely those which the equal sides subtend ; [1. 4] therefore the angle ACB\% equal to the angle CBD. And, since the straight line BC falling on the two straight lines AC, BD has made the alternate angles equal to one another, 25 ^C is parallel to BD. [i. 27] And it was also proved equal to it. Therefore etc. q. e. d. I. joining... (at the extremities which are) in the same directions (respectively). I have for clearness' sake inserted the words in brackets though they are not in Uie original Greek, which has "joining... in the same directions" or "on the same sides,*' heX rd odrd iiipf^ iwi^€vyw6ciuffai. The expression "towards the same parts,'* though usage has sanctioned it, b perhaps not quite satisfiurtory. 15. DC, CB and 18. DCB. The Greek has **BC, CD*' and " BCD" in these places respectively. Euclid is not always careful to write in corresponding order the letters denoting corresponding points in congruent figures. On the contrary, he evidently prefers the alpha- betical order, and seems to disdain to alter it for the sake of bqginners or others who might be confused by it. In the case of angles alteration is perhaps unnecessary ; but in the case ^ of trianeles and pairs of corresponding sides I have ventured to alter the order to that which the mathematician of to-day expects. This proposition is, as Proclus says (p. 385, 5), the connecting link between the exposition of the theory of parallels and the investigation of parallelograms. For, while it only speaks of equal and parallel straight lines connecting those ends of equal and parallel straight lines which are in the same directions, it gives, without expressing the fact, the construction or origin of the parallelogram, so that in the next proposition Euclid is able to speak of " parallel6grammic areas" without any further explanation. Proposition 34. In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas. Let ACDB be a paFallelogrammic area, and BC its diameter ; 21 — 2 324 BOOK I [i. 34 5 I say that the opposite sides and angles of the parallelogram ACDB are equal to one another, and the diameter BC bisects it For, since AB is parallel to CD, and the straight line BC has fallen ioupon them, the alternate angles ABC, BCD are equal to one another. [i. 29] Again, since -<4C is parallel to BD, and BC has fallen upon them, 15 the alternate angles ACB, CBD are equal to one another. [l 29] Therefore ABC, DCB are two triangles having the two angles ABC, BCA equal to the two angles DCB, CBD respectively, and one side equal to one side, namely that 20 adjoining the equal angles and common to both of them, BC\ therefore they will also have the remaining sides equal to the remaining sides respectively, and the remaining angle to the remaining angle ; [i. 26] therefore the side AB is equal to CD, 25 and^Cto^A and further the angle BAC is equal to the angle CDB. And, since the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, the whole an^^le ABD is equal to the whole angle A CD. [C.N.2\ 30 And the angle BAC^^s also proved equal to the angle CDB. ^ Therefore in parallelogrammic areas the opposite sides and angles are equal to one another. I say, next, that the diameter also bisects the areas. For, since AB is equal to CD, 35 and BC is common, the two sides AB, BC are equal to the two sides DC, CB respectively ; and the angle ABC is equal to the angle BCD ; therefore the base AC is also equal to DB, 40 and the triangle ABC is equal to the triangle DCB. [i. 4] Therefore the diameter BC bisects the parallelogram ACDB. Q. E. D. 1.34] PROPOSITION 34 325 I. It b to be observed that, when parallelograms have to be mentioned for the first time, Euclid calls them *'parallelogrammtc areas*' or, more exactly, ''parallelogram*' areas {wapaXKii\6yp€ifA/ia x^^pfa). The meaning is simply areas bounded bv parallel straight lines with the further limitation placed upon the term by Euclid that oiAy jeur-Hded figures are so called, although of course there are certain regular polygons which have opposite sides parallel, and which therefore might be said to be areas bouiMed by parallel straight lines. We Sther from Proclus (p. 30^) that the word "parallelogram** was first introduced by Euclid, It its use was suggestea by i. 33, and that the formation of the word «-a/>aXXiyX(&Y/N^(/Aot (paralleMined) was analogous to that of M^rfpanitw (straight-lined or rectilineal). 17, 18, 40. DCB and 36. DC, CB. The Greek has in these places ** BCD'* and ••C/>, BC respectively. Cf. note on i. 33, lines 15, 18. After specif^ng the particular kinds of parallelograms (squares and rhombi) in which the diagonals bisect the angles which they join, as well as the areas, and those (rectangles and rhomboids) in which the diagonals do not bisect the angles, Proclus proceeds (pp. 390 sqq.) to analyse this proposition with reference to the distinction in Aristotle's AncU. Post (i. 4, 5, 73 a 21 — 74 b 4) between attributes which are only predicable of every individual thing (icara iroi^os) in a class and those which are true of \i primarily {tovtov xpc^ov) and genercUly (ko^oXov). We are apt, says Aristotle, to mistake a proof «aTa vavros for a proof rovrov xpcorov KoBokxn) because it is either impossible to find a higher generality to comprehend all the particulars of which the predicate is true, or to find a name for it (Part of this passage of Aristotle has been quoted above in the note on i. 32, pp. 319 — 320.) Now, says Proclus, adapting Aristotle's distinction to theorems^ the present proposition exhibits the distinction between theorems which are general and theorems which are not general. According to Proclus, the first part of the proposition stating that the opposite sides and angles of a parallelogram are equal is general because the property is only true of parallelograms ; but the second part which asserts that the diameter bisects the area is not general because it does not include all the figures of which this property is true, e.g. circles and ellipses. Indeed, says Proclus, the first attempts upon problems seem usually to have been of this partial character (/Acpucoircpai), and generality was only attained by degrees. Thus ''the ancients, after investigating the fact that the diameter bisects an ellipse, a circle, and a parallelogram respectively, proceeded to investigate what was common to these cases," though " it is difficult to show what is common to an ellipse, a circle and a parallelogram." I doubt whether the supposed distinction between the two parts of the proposition, in point of " generality," can be sustained. Proclus himself admits that it is presupposed that the subject of the proposition is a quadrilateral^ because there are other figures (e.g. regular polygons of an even number of sides) besides parallelograms which have their opposite sides and angles equal; therefore the second part of the theorem is, m this respect^ no more < general than the other, and, if we are entitled to the tacit limitation of the 1 theorem to quadrilaterals in one part, we are equally entitled to it in the other. It would almost appear as though Proclus had drawn, the distinction for the mere purpose of alluding to investigations by Greek geometers on the general subject of diameters of all sorts of figures; and it may have been these which brought the subject to the point at which Apollonius could say in the first definitions at the beginning of his Conies that '* In any bent line^ such as is in one plane, I give the name diameter to any straight line which, being drawn from the bent line, bisects all the straight lines (chords) drawn in the line parallel to any straight line.** The term bent line {KOfLwvktf ypa^'q) includes, e.g. in Archimedes, not only curves, but any composite line made 3«6 BOOK I [>• 34, 35 up of straight lines and curves joined together in any manner. It is of course clear that either diagonal of a parallelogram bisects all lines drawn within the parallelogram parallel to the omer diagonal An-Nairizi gives after i. 31 a neat construction for dividing a straight line into any number of equal parts (ed. Curtze, p. 74, ed. B^thom-Heibeig, pp. 141 — 3) which requires only one measuranent repeated, together with the properties of parallel lines including i. 33, 34. As i. 33, 34 are assumed, I place the problem here. The particular case taken is the problem of dividing a straight line into three equal parts. Let AB be the given straight line. Draw AC^ BD at right angles to it on opposite sides. An-Nairizi takes AC^ BD of the same length and then bisects AC at E and BD at F, But of course it is even simpler to measure AE^ EC along one perpendicular equal and of any length, and BF^ FD along the other also equal and of the same length. Join ED, CF meeting AB \n G^ H respectively. Then shall AG, GH, iET^ all be equal Draw HK pandlel to AC, or at right angles to AB. Since now EC, FD are equal and parallel ED, CFiJt equal and parallel. [i- 33 J And HK was drawn parallel to AC. Therefore ECHK is a parallelogram ; whence KH is equal as well as parallel to EC, and therefore to EA. The triangles EAG, KHG have now two angles respectively equal and the sides AE, /^ equal Thus the triangles are equal in all respects, and AG is equal to GH. Similarly the triangles KHG, FBH are equal in all respects, and GH is equal to HB. If now we wish to extend the problem to the case where AB is to be divided into n parts, we have only to measure (n—\) successive equal lengths along AC and {n-i) successive lengths, each equal to the others, along BD. Then join the first point arrived at' on AC to the last point on BD, the second on ^^C to the last but one on BD, and so on ; and the joining lines cut AB in points dividing it into n equal piauts. Proposition 35. Parallelograms which are on the same base and in the same parallels are equal to one another. Let ABCD, EBCF be parallelograms on the same base BC and in the same parallels AF, BC ; 5 I say that ABCD is equal to the parallelogram EBCF. For, since ABCD is a parallelogram, AD is equal to BC. [l 34] I. 35] PROPOSITIONS 34, 35 3^7 I For the same reason also I EF is equal to BCy iio so that AD is also equal to EF\ \C. N, i] and DE is common ; ! therefore the whole AE is equal to the whole Z?/^ [C. N. 2] But AB is also equal to DC\ [i. 34] therefore the two sides EA, AB are equal to the two sides }SED, DC respectively, and the angle FDC is equal to the angle EAB, the exterior to the interior ; [i. 29] therefore the base EB is equal 20 to the base FQ and the triangle EAB will be equal to the triangle FDC. [1.4] Let DGE be subtracted from each ; therefore the trapezium ABGD which remains is equal to the trapezium EGCF "which remains. [C N. 3] 25 Let the triangle GBC be added to each ; therefore the whole parallelogram A BCD is equal to the whole parallelogram EBCF. [C N. 2] Therefore etc. Q. E. D. 11. FDC. The text has "/>/-C." 33. Let DOE be subtracted. Euclid speaks of the triangle DGE without any explanation that, in the case which he takes (where AD^ EF have no point in common), .^ B£t CD must meet at a point G between the two parallels. He allows tnis to appear from the figure simply. Equality in a new sense. It is important to observe that we are in this proposition introduced for the first time to a new conception of equality between figures. Hitherto we have had equality in the sense of congruence only, as applied to straight lines, angles, and even triangles (cf. i. 4). Now, without any explicit reference to any change in the meaning of the term, figures are inferred to be equal which are equal in area or in content but need not be of the same farm. No definition of equality is anywhere given by Euclid ; we are left to infer its meaning from the few axioms about " equal things." It will be observed that in the above proof the " equality " of two parallelograms on the same base and between the same parallels is inferred by the successive steps (i) of subtracting one and the same area (the triangle DGE) from two areas equal in the sense of congruence (the triangles AEB^ ^PC\ and inferring that the remainders (the trapezia ABGD^ EGCF) are "equal"; (a) of addi^ one and 328 BOOK I [i. 3S the same area (the triangle GBC) to each of the latter ** equal" trapezia, and inferring the equality of the respective sums (the two given parallelograms). As is well known, Simson (after Qairaut) slightly altered the proof in order to make it applicable to all the three possible cases. The alteration substituted one step of subtracting congruent areas (the triangles AEB^ DFC) from one and the same area (the trapezium ABCF) for the iu>o steps above shown of first subtracting and then adding a certain area. While, in either case, nothing more is explicitly used than the axioms that, if equals be added to equals^ the wholes are equal and that, if equals be subtracted from equals^ the remainders are equals there is the further tacit assumption that it is indifferent to what part or from what part of the same or equal areas the same or equal areas are added or subtracted. De Morgan observes that the postulate "an area taken from an area leaves the same area from whatever part it may be taken " is particularly important as the key to equality of non- rectilineal areas which could not be cut into coincidence geometrically. Legendre introduced the word equivalent to express this wider sense of equality, restricting the term equal to things equal in the sense of congruent ; and this distinction has been found convenient. I do not think it necessary, nor have I the space, to give any account of the recent developments of the theory of equivalence on new lines represented by the researches of W. Bolyai, Duhamel, ])e Zolt, Stolz, Schur, Veronese, Hilbert and others, and must refer the reader to Ugo Amaldi's article Sulla teoria dell' equivalenza in Questiani riguardanti la geometria eUmentare (Bologna, 1900), pp. 103 — 142, and to Max Simon, Ober die Entwicklung der Elcmentar-^ometrie im XIX. fahrhumUrt {Ijtvpiagy 1906), pp. 115 — lao, with their full references to the literature of the subject I may however rder to the suggestive distinction of phraseology used by Hilbert (Grundlagen der Geometrie, pp. 39, 40) : (i) "Two polygons are called divisibly-equal (zerlegungsgleich) if they can be divided into z, finite number of triangles which are congruent two and two." (2) "Two polygons are called equal in content {inhaltsgleich) or of equal content if it is possible to add divisibly-equal polygons to them in such a way that the two combined polygons are ditnsibly-equaiy (Amaldi suggests as alternatives for the terms in (i) and (2) the expressions equivalent by sum and equivalent by difference respectively.) From these definitions it follows that " by combining divisibly-equal ' polygons we again arrive at divisibly-equal polygons; and, if we subtract divisibly-equal polygons from divisibly-equal polygons, the polygons remaining are equal in content" The proposition also follows without difficulty that, "if two polygons are divisibly-equal to a third polygon, they are also divisibly-equal to one another ; and, if two polygons are equal in content to a third polygon, they are equal in content to one another." The difTerent cases. As usual, Proclus (pp. 399 — 400), observing that Euclid has given only the most difficult of the three possible cases, adds the other two with separate proofs. In the case where E ii) the figure of the proposition falls between A and A he adds the congruent triangles ABE^ DCF respectively to the smaller trapezium EBCD^ instead of subtracting them (as Simson does) from the larger trapezium ABCF, 1.35] PROPOSITION 35 339 An ancient ** Budget of Paradoxes." Proclus observes (p. 396, 12 sqq.) that the present theorem and the similar one relating to triangles are among the so^^alled iMiradoxical theorems of mathematics, since the uninstructed might well regard it as impossible that the area of the parallelograms should remain the same while the length of the sides other than the base and the side opposite to it may increase indefinitely. He adds that mathematicians had made a collection of such paradoxes, the so-called treasury of paradoxes (6 irofxiSo^os rimoi) — cf. the similar expressions fifWQii dvaXv6fi€voi (treasury of analysis) and rotros d/(rrpovofMVfitvo9 — in the same way as the Stoics with their illustrations {Zavtp ol awo rSj$ Sroas hrl rmv &cy/iar«t>v). It may be that this treasury of paradoxes was the work of Erycinus quoted by Pappus (iii. p. 107, 8) and mentioned above (note on I. 31, p. 290). Locus-theorems and loci in Greek geometry. The proposition i. 35 is, says Proclus (pp. 394 — 6), the first locus-theorem (rorucov ^coipi^/ia) given by Euclid. Accordingly it is in his note on this proposition that Proclus gives us his view of the nature of a locus-theorem and of the meaning of the word locus (roVos) ; and great importance attaches to his words because he is one of the three writers (Pappus and Eutocius being the two others) upon whom we have to rely for all that is known of the Gredc conception of geometrical loci. Proclus' explanation (pp. 304, 15 — 395, 2) is as follows. "I call those (theorems) locus-theorems (roimra) in which the same property is found to exist on the whole of some locus (irpoc oh^ rm, r6iw^\ and (I call) a locus a position of a line or a surface producing one and the same property (ypofifi^ 1j iwi- ^vtCoLS $€<nv voiovcrav Iv icai ravrov crv/utima/ia). *For, of locus-theorems, some are constructed on lines and others on siirfaces (rcSv yap rmnKC^v r& /ack k(m wph% ypafjLfjLOii (rvFurra/uicva, ra Sc irpoc ciri^avciatf ). And, since some lines are plane (cirivcSoi) and others solid (arcpcoi)— those being plane which are simply conceived of in a plane («Sv cv mirc&p 6,irkfj 17 vArftm), and those solid the origin of which is revealed from some section of a solid figure, as the cylin- drical helix and the conic lines (cos rifs tcvXivSpucff^ IXucoi koI ron^ KWfucQi^ ypafifjLwv) — I should say (^rfv av) further that, of locus-theorems on lines, . some give a plane locus and others a solid locus." Leaving out of sight for the moment the class of loci on surfaces^ we find that the distinction between plane and solid loci^ or plane and solid lineSy was similarly understood by Eutocius, who says (Apollonius, ed. Heiberg, 11. p. 184) that ^^ solid loci have obtained their name from the fact that the lines used in the solution of problems regarding them have their origin in the section of solids, for example the sections of the cone and several others.** Similarly we gather from Pappus that plane loci were straight lines and circles, and solid loci were conies. Thus he tells us (vii. p. 672,' 20) that Aristaeus wrote five books of Solid Loci '' supplementary to (literally, continuous with) the conies"; and, though Hultsch brackets the passage (vii. p. 662, 10 — 15) which says plainly that plane loci are straight lines and circles, while solid loci are sections of cones, i.e. parabolas, ellipses and hyperbolas, we have the exactly conesponding distinction drawn by Pappus (in. p. 54, 7 — 16) between plane and solid problems^ plane problems being those solved by means of straight lines and circumferences of circles, and solid problems those solved by means of one or more of the sections of the cone. But, whereas Proclus 330 BOOK I [l3S and Eutocius speak of other solid loci besides conies, there is nothing in Pappus to support the wider application of the term. According to Pappus (ill. p. ^4, i6— ai) problems which could not be solved by means of straight lines, curdes, or conies were linear (yftofAfiuci) because they used for their construction lines having a more complicated and unnatural origin than thoae mentioned, namely such curves as quadrairias^ conchoids and dssoids. Similarly, in the passage supposed to be interpolated, linear loci are distin- guished as those which are neither straight lines nor drdes nor any of the conic sections (vii. p. 662, 13 — 15). Thus the classification given by Produs and Eutodus is less predse than that which we find in Piippus; and the indusion by Proclus of the cylindrical helix among solid loci, on the ground that it arises from a section of a solid figure, would seem to be, in any case^ due to some misapprehension. Comparing these passages and the hints in Pappus about loci on suffaoes (roiroi vpo« iwi^vtUf) with special refere